# Single-index mixture cure model
under monotonicity constraints

Eni<EMAIL_ADDRESS>[ Tsz Pang<EMAIL_ADDRESS>[
Korteweg-de Vries Institute for Mathematics, University of Amsterdam,
Netherlands

###### Abstract

We consider survival data in the presence of a cure fraction, meaning that
some subjects will never experience the event of interest. We assume a mixture
cure model consisting of two sub-models: one for the probability of being
uncured (incidence) and one for the survival of the uncured subjects
(latency). Various approaches, ranging from parametric to nonparametric, have
been used to model the effect of covariates on the incidence, with the
logistic model being the most common one. We propose a monotone single-index
model for the incidence and introduce a new estimation method that is based on
the profile maximum likelihood approach and techniques from isotonic
regression. The monotone single-index structure relaxes the parametric
logistic assumption while maintaining interpretability of the regression
coefficients. We investigate the consistency of the proposed estimator and
show through a simulation study that, when the monotonicity assumption is
satisfied, it performs better compared to the non-constrained single-index/Cox
mixture cure model. To illustrate its practical use, we use the new method to
study melanoma cancer survival data.

survival analysis,

mixture cure model,

single-index model,

isotonic estimation,

kernel smoothing,

###### keywords:

and

##  1 Introduction

Modelling time-to-event data in the presence of subjects that will never
experience the event of interest has gained popularity over the recent
decades. For instance, the advancement of cancer treatments has led to a
larger fraction of patients being cured of their diseases [17]. Another
example can be found in fertility studies [32], where one is interested in the
time to pregnancy while there are infertile couples for whom natural
conception is impossible. Cure models have also been utilized in credit
scoring to model the time to default of a loan applicant and default does not
occur for the majority of debtors [8]. In all these scenarios, the subjects
that are immune to the event of interest are referred to as ‘cured’ (non
susceptible). There are two types of cure models: mixture cure models and
promotion time models. We refer the reader to [1] and [26] for a comprehensive
review of these models.

Mixture cure models assume that the population is a mixture of cured and
susceptible subjects and consist of two sub-models: one for the uncured
probability (incidence) and one for the conditional survival function of the
susceptibles (latency). Initially, fully parametric models with logistic
regression assumption for the incidence and different parametric distributions
for the latency were proposed [9, 10]. Later on, extensions to semi-parametric
models for the latency, such as the Cox proportional hazard (PH) model [25,
28] and the accelerated failure time model [18, 34], were introduced. For the
incidence, a nonparametric estimator was developed in [33] based on the Beran
estimator for the conditional survival function. However, such method is
problematic for multivariate covariates since it requires multi-dimensional
smoothing. To circumvent the curse-of-dimensionality, a single-index model for
the incidence was introduced in [2], linking a linear predictor (index) to the
incidence probability via an unspecified link function. The index achieves
dimension reduction and alleviates the dimensionality issues when estimating
the link function nonparametrically. In certain contexts, the link function is
expected to be monotone, meaning that the cure probability increases/decreases
as the risk score of an individual, given by the index, increases. The widely
used logistic model for the incidence is in particular a monotone single-index
model. In such cases, it is more appropriate to estimate the link function
under monotonicity constraints, while the estimate proposed in [2] is not
guaranteed to be monotone. Another advantage of a monotone link function is
interpretability since the sign of the coefficients of the index tell us
whether a given covariate increases or decreases the cure chances. This
motivates us to investigate estimation of a single-index model for the
incidence under monotonicity constraints. We focus mainly on the incidence
component and assume a Cox PH model for the latency.

Single-index models have been thriving because of their flexibility over
linear and parametric models, while avoiding dimensionality problems of
general nonparametric models. The monotonicity of the link function appears in
numerous applications leading to the popularity of generalized linear models.
Therefore, it is appealing to impose monotonicity constraint on the link
function, which leads to monotone single-index models and has recently become
an active research area [5, 11, 3, 4]. It is also worth mentioning that the
binary choice model in econometrics and the current status linear regression
model [11] are special cases of the monotone single-index model. More in
general, there has been a growing interest in statistical inference under
shape-constraints, such as monotonicity, convexity, log-concavity, etc., which
arise naturally in a wide range of applications [13]. One advantage of such
methods is that they allow for nonparametric estimation without using tuning
parameters. However, a combination of smoothing and shape-constrained
estimation often leads to better finite sample performance [19, 20].

In this paper we introduce an estimation method for the monotone single-index
mixture cure model that is based on the profile maximum likelihood principle
and techniques from isotonic regression in combination with kernel smoothing.
Despite the fact that the monotone single-index model has already been studied
in the literature, its use within the mixture cure model has some unique
features that make the problem more challenging (see Section 3 for a more
detailed discussion). First, in contrast to the standard monotone single-index
models, including the current status linear regression model, where the
response is directly observed, the cure status in mixture cure model is
unknown for the censored subjects. As a result, the likelihood has a more
complicated expression and iterative procedures such as the EM algorithm are
required to solve the optimization problem. Secondly, apart from the
coefficients of the index and the link function, our model contains additional
parameters (finite and infinite dimensional) because of the extra latency
component. Challenges also arise when studying the theoretical properties of
the estimators, which are established in a less straightforward manner as
compared to the monotone single-index model or the current status model, see
Section 4 for a more detailed discussion. We study consistency of the proposed
estimator, which to our best knowledge has not been investigated even for the
single-index mixture cure model without monotonicity constraints [2]. We
illustrate through simulations that imposing monotonicity improves the
behavior of the estimator and makes it more stable with respect to the choice
of the bandwidth compared to the smooth non-monotone estimator. In particular,
one does not need to use time consuming bandwidth selection procedures since a
simple bandwidth choice performs reasonably well. In addition, we observe that
our decision to incorporate a smoothing step to the isotonic estimation of the
link function is motivated by its improved practical performance for finite
sample sizes.

The paper is organized as follows. Section 2 describes the monotone single-
index mixture cure model and the conditions for model identification. Section
3 introduces the estimation procedure, while Section 4 focuses on establishing
consistency of the estimator. The finite sample properties of the proposed
method are investigated through a simulation study and the results are
reported in Section 5. Finally, an illustration of the practical use through a
study of a medical dataset of melanoma cancer patients is provided in Section
6. The proofs and additional simulation results can be found in the Appendix.
Software in the form of R code is available on the GitHub repository
https://github.com/tp-yuen/msic.

##  2 Model description

Let $T$ be a nonnegative random variable denoting the survival time, i.e time
until occurrence of an event of interest, which can be equal to infinity
indicating the possibility of cure. Under the assumption that the survival
time is subject to random right censoring, we observe the follow-up time
$Y=\min{\left(T,C\right)}$ and the censoring indicator
$\Delta={\mathds{1}}\left\\{T\leq C\right\\}$, where $C$ is the censoring
time. Since the duration of the studies is in practice limited, we assume that
$C$ has bounded support. As a result of censoring, the cure status
$B={\mathds{1}}\left\\{T<\infty\right\\}$ is a latent variable and the cured
subjects cannot be distinguished from the censored uncured ones. Since the
cure probability and the survival time of the uncured do not necessarily
depend on the same predictors, we use two sets of covariates
$\bm{\mathrm{X}}\in[d]$ and $\bm{\mathrm{Z}}\in[q]$ that can possibly be the
same or partially/completely different. We assume that $C$ and $T$ are
conditionally independent given the covariates
$(\bm{\mathrm{X}},\bm{\mathrm{Z}})$, which is a rather standard assumption in
survival analysis. In the mixture cure model the survival function is given by

$S\left(t|\bm{\mathrm{x}},\bm{\mathrm{z}}\right)=\mathbb{P}\left(T>t|\bm{\mathrm{X}}=\bm{\mathrm{x}},\bm{\mathrm{Z}}=\bm{\mathrm{z}}\right)=1-p\left(\bm{\mathrm{x}}\right)+p\left(\bm{\mathrm{x}}\right)S_{u}\left(t|\bm{\mathrm{z}}\right),$
(1)

where
$p\left(\bm{\mathrm{x}}\right)=\mathbb{P}\left(B=1|\bm{\mathrm{X}}=\bm{\mathrm{x}}\right)$
is the conditional uncure probability (incidence) and
$S_{u}\left(t|\bm{\mathrm{z}}\right)$ is the conditional survival function for
the uncured (latency). Note that $S_{u}$ is a proper survival function, while
$\lim_{t\to\infty}S\left(t|\bm{\mathrm{x}},\bm{\mathrm{z}}\right)=1-p\left(\bm{\mathrm{x}}\right)$.
In terms of distribution functions we have
$F(t|\bm{\mathrm{x}},\bm{\mathrm{z}})=p\left(\bm{\mathrm{x}}\right)F_{u}(t|\bm{\mathrm{z}})$.
We consider a monotone single-index model for the incidence component, that is

$p\left(\bm{\mathrm{x}}\right)=\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}}\right),$
(2)

for some unknown regression coefficient $\bm{\mathrm{\gamma}}_{0}\in[d]$ and
an unknown link function $\varphi_{0}$ belonging to the set
$\mathcal{M}:=\left\\{\varphi:\rightarrow\left(0,1\right):\ \varphi~{}\text{is
monotone non-decreasing}\right\\}$. For the latency we assume a Cox
proportional hazard model, i.e.

$S_{u}\left(t|\bm{\mathrm{z}}\right)=\mathbb{P}\left(T>t|B=1,\bm{\mathrm{Z}}=\bm{\mathrm{z}}\right)=\exp\left[-\Lambda_{0}\left(t\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{0}\bm{\mathrm{z}}\right)\right],$
(3)

where $\Lambda_{0}$ denotes the baseline cumulative hazard function and
$\bm{\mathrm{\beta}}_{0}\in[q]$ is a vector of regression parameters. Both
$\Lambda_{0}$ and $\beta_{0}$ are left unspecified.

A crucial issue for both the single-index and the mixture cure model is
identifiability meaning that

$\begin{split}&l(Y,\Delta,X,Z;\varphi,\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda)=l(Y,\Delta,X,Z;\tilde{\varphi},\tilde{\bm{\mathrm{\gamma}}},\tilde{\bm{\mathrm{\beta}}},\tilde{\Lambda})\text{
a.s.}\\\
&\implies\varphi=\tilde{\varphi},\quad\bm{\mathrm{\gamma}}=\tilde{\bm{\mathrm{\gamma}}},\quad\bm{\mathrm{\beta}}=\tilde{\bm{\mathrm{\beta}}},\quad\Lambda=\tilde{\Lambda},\end{split}$

where
$l(Y,\Delta,X,Z;\varphi,\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda)$
denotes the log-likelihood of the model given the parameters. For the single-
index model, the parameters $(\varphi_{0},\bm{\mathrm{\gamma}}_{0})$ are not
identifiable without further restrictions since for any $a\in$ we can define
$\tilde{\varphi}(t)=\varphi_{0}(t/a)$,
$\tilde{\bm{\mathrm{\gamma}}}\bm{\mathrm{=}}a\bm{\mathrm{\gamma}}_{0}$ and
have
$\tilde{\varphi}(\tilde{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})=\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}\bm{\mathrm{x}})$.
For the mixture cure model, the parameters are not identifiable if the follow-
up of the study does not contain the support of the event times since it is
not possible to distinguish the event of being cured from the one of being
uncured with survival time larger than the follow-up of the study. To
guarantee identifiability of the model we require the following set of
assumptions, where $\mathcal{X}$ denotes the support of the covariate
$\bm{\mathrm{X}}$ and
$\mathcal{I}_{\bm{\mathrm{\gamma}}}:=\left\\{{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}:\
\bm{\mathrm{x}}\in\mathcal{X}\right\\}$ denotes the support of the index
${\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}$.

  1. (A1)

    1. (i)

$\varphi_{0}$ is differentiable and not constant on
$\mathcal{I}_{\bm{\mathrm{\gamma}}_{0}}$.

    2. (ii)

The parameter $\bm{\mathrm{\gamma}}_{0}$ does not contain an intercept and it
belongs to the $d-1$ dimensional unit sphere
$\mathcal{S}_{d-1}:=\\{\bm{\mathrm{\gamma}}\in[d]\colon{\left\|{\bm{\mathrm{\gamma}}}\right\|_{2}=1}\\}$
with respect to the Euclidean norm $\left\|{\cdot}\right\|_{2}$.

    3. (iii)

The covariate $\bm{\mathrm{X}}$ contains at least one continuous variable and
the continuous components have a joint probability density function.

    4. (iv)

$\mathcal{X}$ is not contained in a proper linear subspace of $[d]$.

    5. (v)

$\mathcal{I}_{\bm{\mathrm{\gamma}}_{0}}$ is not divided into disjoint
intervals by different values of the discrete components.

  2. (A2)

    1. (i)

$\bm{\mathrm{\beta}}_{0}$ does not have an intercept term.

    2. (ii)

The covariance matrix of $\bm{\mathrm{Z}}$ has full rank.

  3. (A3)

    1. (i)

There exists a cure threshold $\tau_{0}<\infty$ such that $T>\tau_{0}\iff
T=\infty$. Moreover
$\mathbb{P}\left(C>\tau_{0}|\bm{\mathrm{X}},\bm{\mathrm{Z}}\right)>0$ for
almost all $\bm{\mathrm{X}}$ and $\bm{\mathrm{Z}}$.

    2. (ii)

The incidence $p\left(\bm{\mathrm{x}}\right)$ in (2) satisfies
$0<p\left(\bm{\mathrm{x}}\right)<1$ for all $\bm{\mathrm{x}}\in\mathcal{X}$.

Assumptions (A1)-(A3) are almost identical to the ones in [2]. Note that we do
not need the assumption that the sign of the first entry of
$\bm{\mathrm{\gamma}}_{0}$ is fixed because we are fixing the direction of
monotonicity for the link function. Assumption (A1) entails the
identifiability of the monotone single-index model in (2) (Theorem 2.1 in
[14]). Given that the incidence in (2) is identifiable, using the same
argument as in the proof of Proposition 1 of [2], it follows that the
monotone-single-index/Cox mixture cure model is identifiable.

##  3 Estimation method

Assume that we have i.i.d. realizations
$\left(y_{i},\delta_{i},\bm{\mathrm{x}}_{i},\bm{\mathrm{z}}_{i}\right)$,
$i=1,\cdots,n$ of $\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}}\right)$. The
observed likelihood function of the mixture cure model is given by

$\displaystyle\begin{split}L_{n}\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda,\varphi\right)=\prod_{i=1}^{n}&\left[\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)\lambda\left(y_{i}\right)e^{{\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}}e^{-\Lambda\left(y_{i}\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}\right)}\right]^{\delta_{i}}\\\
&\times\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)+\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)e^{-\Lambda\left(y_{i}\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}\right)}\right]^{1-\delta_{i}}.\end{split}$
(4)

When the link function is assumed to be known, the parameters
$(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda)$ are estimated via the
maximum likelihood principle. Here, we treat $\varphi$ as a nuisance (infinite
dimensional) parameter and, for any fixed
$(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda)$, we construct a smooth
monotone estimator of $\varphi$. Finally, we consider a new likelihood
criteria with the plug-in estimator of $\varphi$ and apply the maximum
likelihood method. Hence, the estimation procedure consists of the following
three steps:

  1. 1.

For fixed
$\bm{\mathrm{\theta}}=\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda\right)$,
we estimate the link by

$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}=\operatorname*{arg\,max}_{\varphi\in\mathcal{M}_{\epsilon^{\prime}}}L_{n}\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda,\varphi\right),$
(5)

where
$\mathcal{M}_{\epsilon^{\prime}}=\left\\{\varphi:\rightarrow\left[\epsilon^{\prime},1-\epsilon^{\prime}\right]:\
\varphi~{}\text{is monotone non-decreasing}\right\\}$ and
$\epsilon^{\prime}>0$ is a fixed small constant for a truncation on the
uncured probability. See Remark 3.2 below for a discussion on this truncation
and the choice of $\epsilon^{\prime}$. The estimator
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ is computed using the EM algorithm
and techniques from isotonic estimation as explained in Subsection 3.2 below.
The maximizer is not unique but it is uniquely defined at the points
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{i}$, $i=1,\dots,n$. We consider
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ to be a left-continuous step function
that extends constantly to the entire real line.

  2. 2.

A kernel smoothed version of $\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ is
defined by

$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}^{s}\left(u\right)=\int_{u-h}^{u+h}\frac{1}{h}k\left(\frac{u-t}{h}\right)\hat{\varphi}_{n,\bm{\mathrm{\theta}}}\left(t\right)\mathrm{d}t,$
(6)

where $k$ is a symmetric kernel with bounded support $\left[-1,1\right]$ and
$h$ is a chosen bandwidth. By definition and the monotonicity of
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ it follows that
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}^{s}$ is a smooth non-decreasing
function. We illustrate in Appendix B.4 that this smoothing step indeed
improves the behavior of the estimator. For the bandwidth $h$ we follow a
common choice in the literature of smooth isotonic estimators by taking
$h=rn^{-1/5}$, where $r$ is the range of the observed index
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{X}}$. In Appendix B.3 we investigate the
sensitivity of the estimators with respect to the choice of the bandwidth and
conclude that, despite not being the optimal bandwidth, this is a satisfactory
and quick solution. We do not apply any boundary correction for the kernel
estimator but instead extend the isotonic estimator
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ to be constant outside of the range
of the observed data.

  3. 3.

Using the plug-in approach and the maximum likelihood principle,
$\bm{\mathrm{\theta}}_{0}$ is estimated by

$\hat{\bm{\mathrm{\theta}}}_{n}=\left(\hat{\bm{\mathrm{\gamma}}}_{n},\hat{\bm{\mathrm{\beta}}}_{n},\hat{\Lambda}_{n}\right)=\operatorname*{arg\,max}_{\bm{\mathrm{\theta}}}L_{n}\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda,\hat{\varphi}_{n,\bm{\mathrm{\theta}}}^{s}\right),$
(7)

where the maximization is done over
$\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}$, $\bm{\mathrm{\beta}}\in[q]$ and
non-decreasing positive functions $\Lambda$. The estimator is computed
iteratively using the EM algorithm as explained in Subsection 3.2 below. The
whole estimation procedure is described in Algorithm 2 in Appendix B.6. As in
the standard logistic/Cox mixture cure model, we impose the zero tail
constraint meaning that the observations in the plateau are assumed to be
cured. This corresponds to setting
$S_{u}(t|\bm{\mathrm{z}};\Lambda,\bm{\mathrm{\beta}})=\exp(-\Lambda\left(t\right)\exp({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}))=0$
for $t>y_{(r)}$ where $y_{(r)}$ denotes the largest observed event time.

###  3.1 Comparison with similar problems in the literature

Before explaining the computation of the proposed estimators, we comment on
how our problem and method relate to the existing literature on the standard
monotone single-index model [3, 5, 4, 12], which assumes that
$\mathbb{E}\left[Y\mid\bm{\mathrm{X}}\right]=\psi_{0}(\bm{\mathrm{\alpha}}_{0}^{T}\bm{\mathrm{X}})$
for some unknown $\bm{\mathrm{\alpha}}_{0}$ and monotone link function
$\psi_{0}$. Different methods for estimation of
$(\bm{\mathrm{\alpha}}_{0},\psi_{0})$ have been proposed based on the least-
squares principle and adaptations of it without imposing any smoothness
assumptions. The main idea is the following. For fixed $\bm{\mathrm{\alpha}}$,
one can minimize the least squares criterion
$h_{n}(\psi,\bm{\mathrm{\alpha}})=\frac{1}{n}\sum_{i=1}^{n}\\{Y_{i}-\psi(\bm{\mathrm{\alpha}}^{T}\bm{\mathrm{X}}_{i})\\}^{2}$
with respect to $\psi$ on the class of monotone functions, which gives a
$\bm{\mathrm{\alpha}}$-dependent function $\psi_{n,\bm{\mathrm{\alpha}}}$. In
a second step, the function
$h_{n}(\psi_{n,\bm{\mathrm{\alpha}}},\bm{\mathrm{\alpha}})$ is then minimized
over $\bm{\mathrm{\alpha}}$. This would be the standard profile least squares
estimator. Note that since this criterion function for $\bm{\mathrm{\alpha}}$
is not smooth but piecewise constant, the estimator of $\bm{\mathrm{\alpha}}$
is not unique. Moreover, alternative ways to estimate $\bm{\mathrm{\alpha}}$
in the second step have been proposed by using the score approach and
computing the zero-crossings of
$\frac{1}{n}\sum_{i=1}^{n}\\{Y_{i}-\psi_{n,\bm{\mathrm{\alpha}}}(\bm{\mathrm{\alpha}}^{T}\bm{\mathrm{X}}_{i})\\}\bm{\mathrm{X}}_{i}$
or minimizing its squared norm. If the criterion function was continuous in
$\bm{\mathrm{\alpha}}$, these alternative approaches would result in the same
least squares estimator.

The current status linear regression problem can also be seen as a monotone
single-index model where the link is actually a distribution function [12]. In
that setting estimation can be performed via the maximum likelihood principle,
again by first maximizing the likelihood for a fixed index on the class of
distribution functions and then maximizing with respect to
$\bm{\mathrm{\alpha}}$ or solving score equations [11].

In our setting, the response variable that corresponds to the single-index
model for the incidence is the latent cure status $B$. The fact that $B$ is
not always observed makes the use of the least-squares approach not suitable.
Hence, our method is based on the maximum likelihood principle similarly to
the one for the current status linear regression problem. However, in the
current status model, the response (the current status $\Delta$) is observed
and the only unknown parameters are the index and the link function. In our
model, the presence of additional unknown finite and infinite dimensional
parameters $\bm{\mathrm{\beta}},\Lambda$ makes the estimation problem much
more challenging. In particular, both optimization problems in (5) and (7)
cannot be solved directly but only through iterative procedures such as the EM
algorithm. To the best of our knowledge, this is the first case for which a
maximum likelihood estimator under monotonicity constraints, as in (5), cannot
be characterized explicitly. In the proof of Proposition 3.1 below we comment
that, even if one would try to use the standard techniques from isotonic
estimation to characterize the maximizer as the left derivative of a greatest
convex minorant, would end up with an iterative procedure that is the same as
the EM algorithm. In addition, we include a smoothing step which leads to a
smooth monotone estimator of the link function and a continuous criterion for
estimation of $\bm{\mathrm{\theta}}$ in the next step. In Appendix B.4 we
illustrate that smoothing indeed improves the performance of the estimator
compared to the monotone (non-smooth) estimator. As in our case, a truncation
is also needed in the current status regression problem in order to avoid the
link function from being close to 0 and 1. However, since for that setting the
link is a distribution function which necessarily obtains the values 0 and 1,
the truncation is imposed for the likelihood criterion excluding the extreme
observations. For our model, given the assumption (A3)(ii), it seems easier
and more reasonable to restrict to link functions that are bounded away from 0
and 1.

###  3.2 Computation of the estimators

Unlike the Cox proportional hazard model [6], for which the regression
coefficients $\bm{\mathrm{\beta}}$ can be estimated using a profile likelihood
approach independently of the baseline cumulative hazard function $\Lambda$,
the mixture cure model does not possess a likelihood function that can take
advantage of such approach due to the latent uncure status $B$. The
maximization problems in (5) and (7) are solved via the expectation-
maximization (EM) algorithm as in the standard logistic/Cox mixture cure model
[28].

#### EM Algorithm

The uncure status $B_{i}$ of the i-th subject is $B_{i}=1$ (uncured) if
$\delta_{i}=1$ and is unknown otherwise. Given the observed data
$\bm{\mathrm{v}}_{i}=\left(y_{i},\delta_{i},\bm{\mathrm{x}}_{i},\bm{\mathrm{z}}_{i}\right)$,
$i=1,\cdots,n$, the complete-data likelihood function is

$\displaystyle\begin{split}L_{nc}\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda,\varphi\right)=&\prod_{i=1}^{n}\varphi({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})^{B_{i}}[1-\varphi({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})]^{1-B_{i}}\\\
&\times\prod_{i=1}^{n}\left[\lambda\left(y_{i}\right)e^{{\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}}e^{-\Lambda\left(y_{i}\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}\right)}\right]^{\delta_{i}B_{i}}\\\
&\times\prod_{i=1}^{n}\left[e^{-\Lambda\left(y_{i}\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i}\right)}\right]^{(1-\delta_{i})B_{i}}.\end{split}$
(8)

In the $(k+1)$-th iteration of the EM algorithm the parameters are updated as
follows. The E-step of the EM algorithm computes the conditional expectation
of the complete-data log-likelihood $\log L_{nc}$ with respect to the uncured
status $B$ given the parameters of the previous iteration and the observed
data. Specifically, by virtue of the partially observed nature of $B$ and its
linearity in the complete-data log-likelihood function, the E-step is
equivalent to computing

$\begin{split}w_{i}^{(k)}&=\mathbb{E}^{(k)}[{B_{i}\mid\bm{\mathrm{v}}_{i}}]\\\
&=\delta_{i}+\left(1-\delta_{i}\right)\frac{\varphi^{(k)}({\bm{\mathrm{\gamma}}^{(k)T}}\bm{\mathrm{x}}_{i})S_{u}(y_{i}|\bm{\mathrm{z}}_{i};\bm{\mathrm{\beta}}^{(k)},\Lambda^{(k)})}{1-\varphi^{(k)}({\bm{\mathrm{\gamma}}^{(k)T}}\bm{\mathrm{x}}_{i})+\varphi^{(k)}({\bm{\mathrm{\gamma}}^{(k)T}}\bm{\mathrm{x}}_{i})S_{u}(y_{i}|\bm{\mathrm{z}}_{i};\bm{\mathrm{\beta}}^{(k)},\Lambda^{(k)})},\end{split}$
(9)

where the expectation $\mathbb{E}^{(k)}$ is computed using the parameters
${\bm{\mathrm{\theta}}^{(k)},\varphi^{(k)}}$ and
$S_{u}(\cdot|\cdot;\bm{\mathrm{\beta}},\Lambda)$ is obtained according to (3).
Substituting $B_{i}$ with $w_{i}$ in (8), we obtain the expected complete-data
likelihood

$\begin{split}&\prod_{i=1}^{n}\varphi({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})^{w_{i}}[1-\varphi({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})]^{1-w_{i}}\prod_{i=1}^{n}f_{u}({y_{i}|\bm{\mathrm{z}}_{i}})^{\delta_{i}w_{i}}S_{u}(y_{i}|\bm{\mathrm{z}}_{i})^{(1-\delta_{i})w_{i}}\\\
&=\tilde{L}_{nc}^{1}\left(\bm{\mathrm{\gamma}},\varphi\right)\tilde{L}_{nc}^{2}\left(\bm{\mathrm{\beta}},\Lambda\right),\end{split}$
(10)

where, to simplify the notation, we have denoted $w_{i}^{(k)}$ by just
$w_{i}$. The M-step of the algorithm consists in maximizing the expected
complete-data likelihood with respect to the parameters of interest.
Specifically, in (5), we maximize with respect to $\varphi$ over
$\mathcal{M}_{\epsilon^{\prime}}$ while keeping
$\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda\right)$ fixed in all
iterations, while in (7) we maximize over
$\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda\right)$ for a given
$\varphi=\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$. To simplify the
maximization problem in (7), the link function could be kept fixed, equal to
the one obtained in the previous iteration of the EM algorithm with
$\bm{\mathrm{\theta}}=\bm{\mathrm{\theta}}^{(m-1)}$. However, we observe that
in some cases such procedure does not behave well computationally since the
link estimator is very sensitive to the estimator of $\bm{\mathrm{\gamma}}$.
On the other hand, the link estimate seems to be stable with respect to small
changes of the latency parameters $\bm{\mathrm{\beta}}$ and $\Lambda$ from one
iteration to the other. Hence, we only fix $\bm{\mathrm{\beta}}$ and $\Lambda$
as the estimates of the previous iteration and allow
$\varphi=\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ with
$\bm{\mathrm{\theta}}=(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}}^{(m-1)},\Lambda^{(m-1)})$
to still depend on $\bm{\mathrm{\gamma}}$. From (10), we can see that the
expected complete-data likelihood can be factorized into two parts. The first
part only consists of the parameters of the incidence part, while the second
part contains the parameters of the latency part only. Therefore we can
maximize the likelihood for the two parts separately.

#### Monotone Link Estimator

Algorithm 1 in Appendix B.6 describes the procedure of the EM algorithm for
estimation of the monotone link function in (5). The M-step of the EM
algorithm is equivalent to the following maximization problem for a fixed
$\bm{\mathrm{\gamma}}$

$\operatorname*{maximize}_{\varphi\in\mathcal{M}_{\epsilon^{\prime}}}\sum_{i=1}^{n}\left\\{w_{i}\log\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)+\left(1-w_{i}\right)\log\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)\right]\right\\},$
(11)

which belongs to the class of order-restricted maximum likelihood estimation
problems [27]. In addition to the order restriction, a uniform bound
restriction is imposed to the maximum likelihood estimation (MLE) of $\varphi$
for fixed $\bm{\mathrm{\gamma}}$. Such class of order and uniform bound
restricted problems is studied by [15]. Using results from [27] and [15] we
obtain the following characterization, a proof of which can be found in
Appendix A.

######  Proposition 3.1.

The maximizer in (11) exists, it is not unique but it is uniquely defined at
the ordered points
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(1)}<\dots<\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(n)}$
with correspondent values
$\hat{\varphi}_{i}=\max(\epsilon^{\prime},\min(\tilde{\varphi}_{i},1-\epsilon^{\prime}))$,
where $(\tilde{\varphi}_{1},\dots,\tilde{\varphi}_{n})$ are the left
derivatives of the greatest convex minorant of the cumulative sum diagram

$\left\\{\left(0,0\right),\left(i,\sum_{j=1}^{i}w_{(j)}\right),i=1,\cdots,n\right\\}.$

Here $w_{(j)}$ corresponds to the $j$-th order statistic of
${\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}$, $i=1,\dots,n$ .

######  Remark 3.2.

The truncation of the link function is introduced in order to avoid that it
takes the extreme values $0$ and $1$ which would create theoretical problems
with terms that explode to infinity. When $\epsilon^{\prime}$ is chosen to be
very small (for example of the order $10^{-6}$), in practice there would be
almost no difference between the truncated and not truncated version of the
link function since by construction $\tilde{\varphi}_{i}\in[0,1]$. Truncation
is also compatible with Assumption (A3)(ii) above. When the support of
$\bm{\mathrm{x}}$ is assumed to be bounded, there exists $\epsilon^{\prime}$
such that the true link belongs to $\mathcal{M}_{\epsilon^{\prime}}.$

#### Incidence Regression Parameter Estimator

For the estimation of the incidence regression parameter
$\bm{\mathrm{\gamma}}$ in (7), the M-step of the EM algorithm is equivalent to
the following maximization problem

$\operatorname*{maximize}_{\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}}\sum_{i=1}^{n}{\left\\{w_{i}\log\varphi_{\bm{\mathrm{\gamma}}}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})+\left(1-w_{i}\right)\log\left[1-\varphi_{\bm{\mathrm{\gamma}}}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})\right]\right\\},}$
(12)

where $\varphi_{\bm{\mathrm{\gamma}}}$ denotes the smooth monotone link
estimate $\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ for
$\bm{\mathrm{\theta}}=(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}}^{(m-1)},$
$\Lambda^{(m-1)})$, i.e. the parameters $\bm{\mathrm{\beta}}$, $\Lambda$ are
fixed to the estimates of the previous iteration. Here, we impose the
identifiability constraint, $\left\|{\bm{\mathrm{\gamma}}}\right\|_{2}=1$,
leading to a maximization problem with a nonlinear constraint. This problem
can be solved by the augmented Lagrangian method. Such method first
reformulates the problem as an unconstrained problem by introducing penalty
terms for the equality constraints and solving this unconstrained problem by
some interior-point algorithms. The penalty terms are then updated. These two
steps are repeated until convergence. See Chapter 17 of [24] for more details.

#### Latency estimation

For the latency component, the M-step of the EM algorithm can be performed as
in [28]. Specifically, the estimator $\widehat{\bm{\mathrm{\beta}}}_{n}$ for
$\bm{\mathrm{\beta}}$ is computed using the profile likelihood approach as the
maximizer of

$\prod_{i=1}^{n}\left(\frac{\exp({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{i})}{\sum_{j\in\mathcal{R}_{i}}w_{j}\exp({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}_{j})}\right)^{\delta_{i}},$

where $\mathcal{R}_{i}$ denotes the risk set just before time $y_{i}$. The
nonparametric estimator of $\Lambda$ is given by

$\hat{\Lambda}_{n}\left(t\right)=\sum_{i:y_{i}\leq
t}\frac{d_{i}}{\sum_{j\in\mathcal{R}_{i}}w_{j}\exp({\widehat{\bm{\mathrm{\beta}}}}^{T}_{n}\bm{\mathrm{z}}_{j})},$

where $d_{i}$ denotes the number of events at time $y_{i}$. As suggested in
[29], the conditional survival function $\hat{S}_{u}(t|\bm{\mathrm{z}})$ is
set to zero when $t>y_{(r)}$.

##  4 Asymptotic properties

We start by providing some technical intuition on the approach and illustrate
the additional challenges that we face compared to the existing literature on
the monotone single-index model. We will derive consistency of our estimators
using results from the theory of semiparametric M-estimation. Indeed, our
estimator $\hat{\bm{\mathrm{\theta}}}_{n}$ corresponds to the maximizer of an
empirical criterion function that depends on an infinite dimensional nuisance
parameter $\varphi$. The unknown $\varphi$ is replaced by a nonparametric
estimator $\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ depending on
$\bm{\mathrm{\theta}}=\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda\right).$
Denote the log-likelihood for a single observation by

$l(y,\delta,\bm{\mathrm{x}},\bm{\mathrm{z}};\bm{\mathrm{\theta}},\varphi)=\delta\log
f_{u}(y|\bm{\mathrm{z}})+\delta\log\varphi(\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}})+(1-\delta)\log[1-\varphi(\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}})F_{u}(y|\bm{\mathrm{z}})].$

Then we can write

$\hat{\bm{\mathrm{\theta}}}_{n}=\operatorname*{arg\,max}_{\bm{\mathrm{\theta}}}M_{n}(\bm{\mathrm{\theta}},\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}),\qquad
M_{n}(\bm{\mathrm{\theta}},\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}})=\frac{1}{n}\sum_{i=1}^{n}l\left(y_{i},\delta_{i},\bm{\mathrm{x}}_{i},\bm{\mathrm{z}}_{i};\bm{\mathrm{\theta}},\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}\right).$

We will show in Proposition 4.3 below that such maximizer exists and is
finite. The asymptotic version of $M_{n}$ is given by
$M(\bm{\mathrm{\theta}},\varphi_{\bm{\mathrm{\theta}}})=\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}},\varphi_{\bm{\mathrm{\theta}}}\right)\right],$
where the infinite dimensional nuisance parameter is allowed to depend on
$\bm{\mathrm{\theta}}$. On the other hand, for fixed
$\bm{\mathrm{\theta}}=\left(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}},\Lambda\right)$,
we define
$l_{\bm{\mathrm{\theta}}}\left(\varphi\right):=\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}},\varphi\right)\right]$
and

$\varphi_{0,\bm{\mathrm{\theta}}}:=\operatorname*{arg\,max}_{\varphi\in\mathcal{M}_{\epsilon^{\prime}}}l_{\bm{\mathrm{\theta}}}\left(\varphi\right),$
(13)

where
$\mathcal{M}_{\epsilon^{\prime}}:=\left\\{\varphi:\rightarrow\left[\epsilon^{\prime},1-\epsilon^{\prime}\right]:\
\varphi~{}\text{is monotone non-decreasing}\right\\}$. This corresponds to the
asymptotic version of the estimator $\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$
in (5). Proposition 4.1 below guarantees that the maximizer
$\varphi_{0,\bm{\mathrm{\theta}}}$ exists and is unique. Note that from
Assumptions (A1)(A1)(i) and (A4) below, it follows that there exists
$\epsilon>0$ such that
$\epsilon\leq\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}\bm{\mathrm{x}})\leq
1-\epsilon$ for all $\bm{\mathrm{x}}\in\mathcal{X}$. We assume that
$\epsilon^{\prime}$ in the definition of
$\hat{\varphi}_{n,\bm{\mathrm{\theta}}}$ is chosen such that
$\epsilon^{\prime}<\epsilon$. In this way,
$\varphi_{0}\in\mathcal{M}_{\epsilon^{\prime}}$ and in particular we have
that, if $\bm{\mathrm{\theta}}=\bm{\mathrm{\theta}}_{0}$, then
$\varphi_{0,\bm{\mathrm{\theta}}}=\varphi_{0}$ (Proposition 4.2). We will show
that
$\bm{\mathrm{\theta}}_{0}=\operatorname*{arg\,max}_{\bm{\mathrm{\theta}}}M(\bm{\mathrm{\theta}},\varphi_{0,\bm{\mathrm{\theta}}}),$
which is the foundation behind the estimation strategy. Then, to obtain
consistency of of the estimator for $\bm{\mathrm{\theta}}_{0}$ we will check
the conditions of Theorem 1 in [7]. Specifically we need that
$\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ is a consistent estimator for
$\varphi_{0,\bm{\mathrm{\theta}}}$ uniformly over $\bm{\mathrm{\theta}}$, the
empirical criterion function $M_{n}(\bm{\mathrm{\theta}},\varphi)$ is a good
approximation of the asymptotic criterion $M(\bm{\mathrm{\theta}},\varphi)$
uniformly over $\bm{\mathrm{\theta}}$ and $\varphi$, the function
$M(\bm{\mathrm{\theta}},\varphi)$ is continuous with respect to $\varphi$ at
$\varphi_{0,\bm{\mathrm{\theta}}}$ uniformly over $\bm{\mathrm{\theta}}$. Such
conditions will be proved in Theorem 4.7.

The function $\varphi_{0,\bm{\mathrm{\theta}}}$ defined in (13) plays a
fundamental role in the theoretical analysis of our estimators and the main
challenges we face arise from the fact that we do not have an explicit
characterization of this function. The counterpart of this function in the
monotone single-index model is
$\psi_{\bm{\mathrm{\alpha}}}(u)=\mathbb{E}[\psi_{0}(\bm{\mathrm{\alpha}}_{0}^{T}\bm{\mathrm{X}})\mid\bm{\mathrm{\alpha}}^{T}\bm{\mathrm{X}}=u]$
(see for example equation (5) in [4]), while in the current status model is
$F_{\bm{\mathrm{\beta}}}(u)=\mathbb{E}[F_{0}(T-\bm{\mathrm{\beta}}_{0}^{T}\bm{\mathrm{}})\mid
T-\bm{\mathrm{\beta}}^{T}\bm{\mathrm{X}}=u]$ (see equation 3.2 in [11]). In
both cases, this function can be seen as the expected value of the true
single-index model when we fix the index to a given value. Having this
explicit characterization makes it easier to deal with this function and in
particular, properties such as continuity, differentiability of
$\psi_{\bm{\mathrm{\alpha}}}$ or $F_{\bm{\mathrm{\beta}}}$ can be derived from
assumptions on the true link function. In our case, we can characterize
$\varphi_{0,\bm{\mathrm{\theta}}}$ as

$\varphi_{0,\bm{\mathrm{\theta}}}(u)=\mathbb{E}\left[\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}\bm{\mathrm{X}})\mid\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{X}}=u\right]$
(14)

only if the covariates $\bm{\mathrm{X}}$ and $\bm{\mathrm{Z}}$ are
independent, censoring is independent of all the other variables and only for
$\bm{\mathrm{\theta}}=(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}}_{0},\bm{\mathrm{\Lambda}}_{0})$
(see details in Appendix A). However, such assumptions are too strong for
practical purposes since it is quite common in particular to have
$\bm{\mathrm{X}}=\bm{\mathrm{Z}}$. Hence, we prefer not to restrict ourselves
to such scenario. Without such characterization, even just arguing continuity
of $\varphi_{0,\bm{\mathrm{\theta}}}$ is quite challenging and technical (see
Proposition 4.4) and requires assumptions that are more difficult to interpret
such as assumption (A8) below. For a fixed $\bm{\mathrm{\theta}}$, the smooth
kernel estimator $\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ is an estimator
of $\varphi_{0,\bm{\mathrm{\theta}}}$, hence in order to obtain the rate of
convergence of the estimators, one would need also (twice) differentiability
of $\varphi_{0,\bm{\mathrm{\theta}}}$. Given the technicalities of proving by
contradiction that $\varphi_{0,\bm{\mathrm{\theta}}}$ is continuous, we do not
explore this direction further.

The second challenge arises from the fact that our parameter
$\bm{\mathrm{\theta}}$ contains not only the index $\bm{\mathrm{\gamma}}$ but
also the latency parameters $\bm{\mathrm{\beta}}$, $\Lambda$. In particular,
$\bm{\mathrm{\theta}}$ is not finite dimensional. The theory of semiparametric
M-estimators in [7] allows $\bm{\mathrm{\theta}}$ to be infinite dimensional
only for the consistency part but obtaining the rate of convergence requires
an Euclidean parameter $\bm{\mathrm{\theta}}$. Hence, one would first need to
extend the standard results of semiparametric M-estimation to this scenario.
Note that obtaining the limit distribution of the latency parameters
$\hat{\bm{\mathrm{\beta}}}_{n},\hat{\Lambda}_{n}$ is equally challenging since
they are dependent on $\hat{\varphi}^{s}_{n,\bm{\mathrm{\theta}}}$ and
$\hat{\bm{\mathrm{\gamma}}}_{n}$ (see Definition (7)) and cannot be dealt with
separately. Because of these two main issues, within this paper, we focus on
the consistency property of the estimators and leave the rate of convergence
and asymptotic normality to be subject of future research.

The results that we previously described intuitively are formulated rigorously
below. First we list the required assumptions. The identifiability assumptions
(A1)-(A3) are assumed throughout this section. In what follows
$B_{\bm{\mathrm{c}}}(r)$ will denote a closed ball of center $\bm{\mathrm{c}}$
and radius $r$ in a given metric space, which is some $[d]$ with the Eucledian
norm if not specified otherwise.

  1. (A4)

The covariates $\bm{\mathrm{X}}$ and $\bm{\mathrm{Z}}$ have bounded supports
$\mathcal{X}$ and $\mathcal{Z}$ respectively. That is, $\mathcal{X}\subset
B_{\bm{\mathrm{0}}}(r_{1})$ for some $r_{1}>0$ and $\mathcal{Z}\subset
B_{\bm{\mathrm{0}}}(r_{2})$ for some $r_{2}>0$.

  2. (A5)

There exists $\delta_{0}>0$ such that, for all $\bm{\mathrm{\gamma}}\in
B_{\bm{\mathrm{\gamma}}_{0}}(\delta_{0})$,
${\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}$ has a density with respect to the
Lebesgue measure. We denote such density by
$g_{{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}}(\cdot)$.

  3. (A6)

$\bm{\mathrm{\gamma}}_{0}$ and $\bm{\mathrm{\beta}}_{0}$ lie in the interior
of compact sets $\Gamma=\mathcal{S}_{d-1}\cap
B_{\bm{\mathrm{\gamma}}_{0}}(\delta_{0})$ and $\mathcal{B}$ respectively.

  4. (A7)

The function $\Lambda_{0}(t)$ defined on $[0,\tau_{0})$ is non-decreasing and
$\Lambda_{0}(\tau_{0}-)<\infty$. We define
$\Lambda_{0}(\tau_{0})=\Lambda_{0}(\tau_{0}-)$, i.e. extend $\Lambda$ at
$\tau_{0}$ by left-continuity.

Consider
$\bm{\mathrm{\theta}}\in\Theta=\Gamma\times\mathcal{B}\times\mathcal{D}$,
where $\mathcal{D}$ is the space of nondecreasing functions $\Lambda$ on
$[0,\tau_{0}]$ such that $\Lambda(0)=0$ and $\Lambda(\tau_{0})<\infty$.

  1. (A8)

For any $\bm{\mathrm{\theta}}\in\Theta$, the function

$u\longmapsto\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\left(Y|\bm{\mathrm{Z}}\right)}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left(\tilde{u}\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right)}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]$

is continuous for all $\tilde{u}\in\mathcal{I}_{\bm{\mathrm{\gamma}}}$, where
$F_{u,0}\left(Y|\bm{\mathrm{Z}}\right)=1-S_{u}\left(Y|\bm{\mathrm{Z}}\right)$
with the true parameters $\bm{\mathrm{\beta}}_{0}$ and $\Lambda_{0}$ as in
(3), $F_{u}(Y|\bm{\mathrm{Z}})=1-S_{u}(Y|\bm{\mathrm{Z}})$ with parameters
$\bm{\mathrm{\beta}}$ and $\Lambda$.

  2. (A9)

The family of density functions
$\\{g_{{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}}:\bm{\mathrm{\gamma}}\in\Gamma\\}$
is uniformly equicontinuous.

  3. (A10)

The density functions $g_{{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}}$ for
$\bm{\mathrm{\gamma}}\in\Gamma$ and
$g_{{\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{Z}}}$ for
$\bm{\mathrm{\beta}}\in\mathcal{B}$ are uniformly bounded from above by some
positive constants $\bar{q}_{1},$ $\bar{q}_{2}$ respectively.

Assumptions (A4)-(A6) are standard assumptions made also in the standard
single index model and the current status model. Assumption (A7) is a standard
assumption of the mixture cure model, see [21, 23]. Together with assumption
(A3), it essentially means that the distribution of the survival times for the
uncured subjects has a jump at $\tau_{0}$, i.e. there is positive probability
for the event to happen at $\tau_{0}$. This is mainly a technical condition
for consistency of the estimators (see discussion in [23]) and the probability
mass at $\tau_{0}$ can be arbitrarily small, hence reasonable in practice.
Assumption (A8) is needed to guarantee the continuity of
$\varphi_{0,\bm{\mathrm{\theta}}}$ since we do not have an explicit expression
for such function. If there was no latency component $F_{u}$ then this
assumption reduces to continuity of
$\mathbb{E}[\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)\mid{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u]$,
which is standard in the single-index model. Assumptions (A9) and (A10) are
required in order to get uniform consistency of the link estimate. (A9) is for
example satisfied if the density functions
$g_{{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}}\left(\cdot\right)$ are
continuously differentiable on their support
$\mathcal{I}_{\bm{\mathrm{\gamma}}}$ for all $\bm{\mathrm{\gamma}}\in\Gamma$
with uniformly bounded derivative.

######  Proposition 4.1.

Suppose that Assumptions (A4) and (A5) hold. Then, for any
$\bm{\mathrm{\theta}}\in\Theta$, the maximizer
$\varphi_{0,\bm{\mathrm{\theta}}}$ in (13) exists and is unique.

######  Proposition 4.2.

For
$\bm{\mathrm{\theta}}=\left(\bm{\mathrm{\gamma}}_{0},\bm{\mathrm{\beta}}_{0},\Lambda_{0}\right)$,
we have $\varphi_{0,\bm{\mathrm{\theta}}}=\varphi_{0}$.

######  Proposition 4.3.

Suppose that Assumptions (A4) and (A6) hold. Then the maximum likelihood
estimator
$(\hat{\bm{\mathrm{\gamma}}}_{n},\hat{\bm{\mathrm{\beta}}}_{n},\hat{\Lambda}_{n})$
defined as in (7) exists and is finite.

######  Proposition 4.4.

Suppose that Assumptions (A4), (A5) and (A8) hold. Then, for any
$\bm{\mathrm{\theta}}\in\Theta$, the function
$u\longmapsto\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$ is continuous.

As in Lemma 1 of [21], using Assumption (A4), it can be shown that there
exists $M>0$ such that $\sup_{n}\hat{\Lambda}_{n}(\tau_{0})\leq M$ a.s..
Hence, we can restrict to
$\bm{\mathrm{\theta}}\in\tilde{\Theta}=\Gamma\times\mathcal{B}\times\tilde{\mathcal{D}}$,
where $\tilde{\mathcal{D}}$ is the subset of $\mathcal{D}$ consisting of
functions $\Lambda$ that are uniformly bounded by $M$.

######  Proposition 4.5.

Suppose that Assumptions (A4)-(A7) and (A10) hold, then

$\mathbb{P}\left[\lim\limits_{n\rightarrow\infty}\sup_{\bm{\mathrm{\theta}}\in\tilde{\Theta}}\int_{\mathcal{X}}\left\\{\hat{\varphi}_{n,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)\right\\}^{2}\mathrm{d}Q_{\bm{\mathrm{X}}}\left(\bm{\mathrm{x}}\right)=0\right]=1,$

where $Q_{\bm{\mathrm{X}}}(\cdot)$ denotes the distribution function of
$\bm{\mathrm{X}}$.

Let $k$ be a symmetric kernel density function with support
$\left[-1,1\right]$ that satisfies $k\left(x\right)\leq K<\infty$ for all
$x\in\left[-1,1\right]$. To simplify the notation, we define
$k_{h}\left(u\right)=h^{-1}k\left(u/h\right)$. Here $h=h_{n}>0$ is a bandwidth
that depends on the sample size $n$ and satisfies $h_{n}\xrightarrow{}0$ as
$n\xrightarrow{}\infty$. Since within this paper we do not investigate further
the rate of convergence of the estimators, no additional restrictions are
imposed on the bandwidth. However, it is known that the optimal order
bandwidth for estimation of a twice differentiable function is $n^{-1/5}$ and,
if the bandwidth is of order $n^{-1/3}$ or smaller, the smooth estimator is
not much different from the non-smooth one. Hence, in practice we take
$h=rn^{-1/5}$, where $r$ is the range of the index
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{X}}$. This is a common choice in the
literature of smooth isotonic estimators that is simple and behaves well. For
a more detailed investigation of the role of the bandwidth see Appendix B.3.

######  Proposition 4.6.

Suppose that Assumptions (A4)-(A10) hold, then

$\mathbb{P}\left[\lim\limits_{n\rightarrow\infty}\sup_{\bm{\mathrm{\theta}}\in\tilde{\Theta}}\int_{\mathcal{X}}\left\\{\hat{\varphi}_{n,\bm{\mathrm{\theta}}}^{s}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)\right\\}^{2}\mathrm{d}Q_{\bm{\mathrm{X}}}\left(\bm{\mathrm{x}}\right)=0\right]=1.$

Next we show that the estimator
$\hat{\bm{\mathrm{\theta}}}_{n}=(\hat{\bm{\mathrm{\gamma}}}_{n},\hat{\bm{\mathrm{\beta}}}_{n},\hat{\Lambda}_{n})$
of $\bm{\mathrm{\theta}}_{0}$ is weakly consistent.

######  Theorem 4.7.

Suppose that Assumptions (A4)-(A10) hold, then

$\|\hat{\bm{\mathrm{\gamma}}}_{n}-\bm{\mathrm{\gamma}}_{0}\|_{2},\qquad\|\hat{\bm{\mathrm{\beta}}}_{n}-\bm{\mathrm{\beta}}_{0}\|_{2}\qquad\text{and}\qquad\sup_{t\in\left[0,\tau_{0}\right]}|\hat{\Lambda}_{n}\left(t\right)-\Lambda_{0}\left(t\right)|$

converge to zero in probability as $n\to\infty$.

The following corollary shows the consistency of the estimated cure
probabilities and the estimated survival function for the uncured.

######  Corollary 4.8.

Suppose that Assumptions (A4)-(A10) hold. We suppose further that
$\varphi_{0}$ has bounded derivative on $\mathcal{I}_{0}$. Then, for any
$\bm{\mathrm{z}}\in\mathcal{Z}$,

$\int_{\mathcal{X}}\left\\{{\hat{\varphi}}_{n,\hat{\bm{\mathrm{\theta}}}_{n}}^{s}({{\hat{\bm{\mathrm{\gamma}}}}^{T}_{n}\bm{\mathrm{x}}})-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}}\right)\right\\}^{2}\mathrm{d}Q_{\bm{\mathrm{X}}}\left(\bm{\mathrm{x}}\right)\qquad\text{and}\qquad\sup_{t\in\left[0,\tau_{0}\right]}|\hat{S}_{u}(t|\bm{\mathrm{z}})-S_{u}(t|\bm{\mathrm{z}})|$

converge to zero in probability as $n\to\infty$, where
$S_{u}(t|\bm{\mathrm{z}})$ and $\hat{S}_{u}(t|\bm{\mathrm{z}})$ are defined as
in (3) using ${\bm{\mathrm{\beta}}}_{0}$, $\Lambda_{0}$ and the estimated
parameters $\hat{\bm{\mathrm{\beta}}}_{n}$, $\hat{\Lambda}_{n}$ respectively.

##  5 Simulation study

In the simulation study, we consider different settings to evaluate the finite
sample behavior of the estimator proposed in Section 3 and compare it with the
SIC method proposed in [2]. We simulate 500 datasets from the mixture cure
model introduced in Section 2, where
$\bm{\mathrm{X}}={\left(X_{1},\cdots,X_{4}\right)}^{T}$ contains four
independent covariates: $X_{1}\sim U[0,1]$, $X_{2}\sim N(0,1)$, $X_{3}$ and
$X_{4}$ are Bernoulli variables with parameters $0.3$ and $0.6$ respectively,
and $\bm{\mathrm{Z}}={\left(Z_{1},Z_{2}\right)}^{T}$ with $Z_{1}=X_{1}$ and
$Z_{2}=X_{4}$. We consider a Weibull model with parameters $\lambda=1.5$ and
$k=2.2$ for the baseline distribution of the uncured subjects. The random
right censoring time $C$ follows the exponential distribution with rate
$\lambda_{C}$. We consider three experiments A, B and C, with three different
non-decreasing link functions.

${\varphi_{A}(u)=\frac{\exp(c+u)}{1+\exp(c+u)},\,\varphi_{B}(u)=\frac{\exp[\psi(c,u)]}{1+\exp[\psi(c,u)]},\,\varphi_{C}(u)=\frac{1+\tanh(c+u^{3})}{2},}$

where $c$ is an intercept term,
$\psi(c,u)=0.75\Phi\\{\left(c+u\right)+0.5\\}+0.25\Phi\\{0.5\left(c+u\right)^{3}\\}$
and $\Phi$ is the cdf of the standard normal distribution. The first two links
correspond to Scenarios 1 and 2 of the simulation study investigated by [2],
while $\varphi_{C}$ is a scaled hyperbolic tangent function. These three link
functions are considered for exploring the influence of the shape and the
steepness of the true link function to the model estimation performance.
Figure 1 shows the plots of the three link functions over $[-4,4]$ when the
intercept term $c=0$. Among these three link functions, $\varphi_{C}$ is the
steepest and $\varphi_{B}$ is the flattest.

Figure 1: Link functions when the intercept term $c=0$. Table 1: Simulation settings  Expt. | $c$ | ${\gamma}_{01}$ | ${\gamma}_{02}$ | ${\gamma}_{03}$ | ${\gamma}_{04}$ | ${\beta}_{01}$ | ${\beta}_{02}$ | $\lambda_{C}$ |  Cure prop. |  Cens. rate | Plateau
---|---|---|---|---|---|---|---|---|---|---|---
A | 1.2 | -0.2383 | 0.7423 | 0.3156 | 0.5409 | -0.8 | 0.5 | 0.1 | 0.2090 | 0.2674 | 0.1675
0.3 | 0.3691 | 0.1108
B | 0.5 | -0.7826 | 0.4368 | -0.2599 | 0.3594 | -0.6 | 0.8 | 0.1 | 0.3352 | 0.3792 | 0.2734
0.4 | 0.4902 | 0.1554
C | 0.2 | 0.1057 | 0.7899 | -0.4883 | 0.3556 | 0.6 | 0.4 | 0.15 | 0.3912 | 0.4415 | 0.3092
0.5 | 0.5390 | 0.1867

The choices for the parameters $\bm{\mathrm{\gamma}}_{0}$,
$\bm{\mathrm{\beta}}_{0}$, and $\lambda_{C}$ are given in Table 1 as well as
the averages, over the 500 simulated datasets for each setting, of the cure
proportion, the censoring rate and the proportion of observations in the
plateau. $\bm{\mathrm{\gamma}}_{0}$ is chosen such that it has unit Euclidean
norm to ensure model identifiability and it leads to different cure
proportions. For each experiment we also consider two different censoring
scenarios while maintaining a reasonable percentage of observations in the
plateau.

The simulation experiments are carried out with sample size $n$ of $250$ and
$500$. The SIC method proposed by [2] and our estimation method (mSIC) are
applied to each dataset. For comparison we compute the mean squared error
(MSE) of the estimate of the cure probability, bias and variance of the
estimates of both $\bm{\mathrm{\gamma}}$ and $\bm{\mathrm{\beta}}$. The MSE
for the cure probability is defined as

$\text{MSE}(\hat{p},p_{0})=\frac{1}{K}\sum_{k=1}^{K}\\{\hat{\varphi}({\hat{\bm{\mathrm{\gamma}}}}^{T}\bm{\mathrm{x}}_{k})-{\varphi_{0}}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}}_{k})\\}^{2},$

where the summation is over a grid of points. For $x_{k1}$ and $x_{k2}$ we
take a grid of size $0.01$ on $\left[0,1\right]$ and $\left[-3,3\right]$
respectively, while $x_{k3},x_{k4}\in\left\\{0,1\right\\}$. The bias of the
coefficient estimates is the mean of the Euclidean norms of the differences
between the coefficient estimate and the true parameter over 500 replications.
The variance of the coefficient estimates is the sample variance of the
Euclidean norms of the coefficient estimates.

For both methods, we initialize the algorithms as follows. The initial link
estimate is the logistic function, $\hat{\bm{\mathrm{\gamma}}}_{(0)}$ is the
estimate from fitting a logistic regression model to the censoring indicator
against the covariates $\bm{\mathrm{X}}$, initial estimates for the latency
are obtained from fitting the standard Cox model to the uncensored
observations. Both algorithms terminate when the difference of the estimators
from one iteration to the other, in Eucledian norm, is smaller then $10^{-5}$.
Other configurations for the SIC method are set as stated in Section 3 of [2].
Since the SIC method sets $\hat{\gamma}_{1}=\pm 1$ for the model
identifiability condition, at the end we normalize the obtained estimate. For
our method we used the triweight kernel function for smoothing the monotone
link estimate and set the bandwidth parameter at the $k$-th iteration to be
$h_{k}=r_{k}n^{-1/5}$, where $r_{k}$ is the range of the index
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{X}}$ computed at the $k$-th iteration.
This is a common choice for smooth isotonic estimators and behaves well in
practice. We note that the bandwidth choice is less problematic in the case of
smooth monotone estimators than for the standard kernel estimator. A more
detailed investigation of the role of the bandwidth is provided in Appendix
B.3. For the SIC method, cross-validation is used to select the bandwidth but
in our experience that is not stable in practice and the search interval needs
to be chosen depending on the range of the index (see discussion in Appendix
B.5). Regarding the truncation parameter, we set $\epsilon^{\prime}=10^{-6}$
for both experiments A and C. For experiment B, given the small range of
values for the incidence, the estimation is very challenging. Since the
isotonic estimator is known to have problems at the boundary, in these type of
situations it is better to determine the upper and lower truncation in a data-
driven way instead of taking a very small $\epsilon^{\prime}$. We use the
method proposed by [22] for the range-regularized isotonic regression problem.
Details are given in Appendix B.1.

Table 2: Simulation results Expt. | Size | $\lambda_{C}$ | Method | $\text{MSE}\left(\hat{p},p_{0}\right)$ | $\hat{\bm{\mathrm{\gamma}}}$ | $\hat{\bm{\mathrm{\beta}}}$
---|---|---|---|---|---|---
Mean | Variance | Bias | Variance | Bias | Variance
A | 250 | 0.1 | mSIC | 0.00939 | 4.39E-05 | 0.57368 | 0.05038 | 0.28080 | 0.02709
SIC | 0.01688 | 1.41E-04 | 0.71096 | 0.05077 | 0.28151 | 0.02708
0.3 | mSIC | 0.01141 | 5.92E-05 | 0.61280 | 0.05145 | 0.30371 | 0.03408
SIC | 0.01940 | 1.55E-04 | 0.76052 | 0.06128 | 0.30605 | 0.03442
500 | 0.1 | mSIC | 0.00562 | 1.50E-05 | 0.43937 | 0.03810 | 0.19793 | 0.01321
SIC | 0.00967 | 4.57E-05 | 0.58563 | 0.04825 | 0.19799 | 0.01331
0.3 | mSIC | 0.00668 | 2.01E-05 | 0.48043 | 0.03989 | 0.22269 | 0.01667
SIC | 0.01183 | 7.44E-05 | 0.61136 | 0.04907 | 0.22371 | 0.01716
B | 250 | 0.1 | mSIC | 0.00697 | 6.06E-05 | 0.99875 | 0.20072 | 0.30717 | 0.03557
SIC | 0.00831 | 9.70E-05 | 1.02873 | 0.26451 | 0.30798 | 0.03525
0.4 | mSIC | 0.00912 | 9.79E-05 | 0.94890 | 0.18028 | 0.35858 | 0.04353
SIC | 0.01041 | 1.34E-04 | 1.04764 | 0.25388 | 0.35971 | 0.04412
500 | 0.1 | mSIC | 0.00433 | 2.25E-05 | 0.88094 | 0.17064 | 0.21738 | 0.01595
SIC | 0.00504 | 4.17E-05 | 0.91734 | 0.26178 | 0.21715 | 0.01596
0.4 | mSIC | 0.00591 | 3.50E-05 | 0.86886 | 0.17217 | 0.24466 | 0.02036
SIC | 0.00578 | 4.44E-05 | 0.97191 | 0.25303 | 0.24623 | 0.02122
C | 250 | 0.15 | mSIC | 0.00583 | 3.49E-06 | 0.29442 | 0.01686 | 0.32893 | 0.03600
SIC | 0.01474 | 1.79E-04 | 0.40554 | 0.01987 | 0.32966 | 0.03599
0.5 | mSIC | 0.00666 | 6.10E-06 | 0.33582 | 0.02433 | 0.37579 | 0.05071
SIC | 0.01847 | 2.82E-04 | 0.45743 | 0.02754 | 0.38216 | 0.05140
500 | 0.15 | mSIC | 0.00446 | 4.52E-05 | 0.20682 | 0.01075 | 0.22675 | 0.01747
SIC | 0.00811 | 5.61E-05 | 0.32336 | 0.01965 | 0.22658 | 0.01761
0.5 | mSIC | 0.00459 | 1.41E-06 | 0.23880 | 0.01169 | 0.26455 | 0.02287
SIC | 0.01038 | 8.51E-05 | 0.35384 | 0.02201 | 0.26582 | 0.02345

Table 2 summarizes the simulation results, including the MSE of the link
estimates, bias and variance of the coefficient estimates, for both SIC and
mSIC methods. In terms of the MSE for the cure probability, the mSIC method
has lower mean and variance compared to the SIC method among all simulation
settings, except for Experiment B with sample size of $500$ and
$\lambda_{C}=0.4$. This indicates that mSIC performs better in inferring the
incidence and gives less dispersed estimates. The mSIC method behaves better
in estimating $\bm{\mathrm{\gamma}}$ and $\bm{\mathrm{\beta}}$, in terms of
bias and variance among all simulation configurations. As expected, as the
sample size increases the performance of both methods improves, while it
deteriorates when the censoring rate increases.

For experiment B, we find that range-regularization method shrinks the range
of the upper and lower truncation to zero for about one-fifth of the
replications for each simulation settings. This leads to a constant estimate
of the link function which poses a practical identification issue for
$\bm{\mathrm{\gamma}}$ and causes the high variability of
$\hat{\bm{\mathrm{\gamma}}}$. We look separately at the cases with a constant
(non-constant) estimate for the link function and report the correspondent
MSE, bias and variance in Tables 4-5 in Appendix B.1. Results indicate that,
when our method estimates a constant link function, the SIC method behaves no
better than a constant link estimate. Table 5 also suggests mSIC performs
better in estimating the incidence for almost all cases with a non-constant
link estimate. As expected, the variance of $\hat{\bm{\mathrm{\gamma}}}$ for
the cases with a constant estimated link function is higher than that for a
non-constant link. This practical identifiability issue may be a result of the
flatness of the true link function $\varphi_{B}$, although it is not a
constant function. Note that the true incidence probability ranges from around
$0.52$ to around $0.73$. We recommend that the range-regularized method should
be used in practice to determine the upper and lower truncation in order to
identify situations for which there is risk of practical identifiability as in
experiment B. In such cases, one should be careful in interpreting the
results.

We note that if the true link function is not monotone, using mSIC would lead
to model mispecification and SIC would be preferred over mSIC. We illustrate
this through an additional simulation setting in Appendix B.2. However, we
expect that for small deviations from monotonicity, mSIC would still provide
reasonable estimates. Finally, we also investigated the role of the additional
smoothing step 2 in the estimation procedure described in Section 3. Results
in Appendix B.4 illustrate that the smooth monotone estimator behaves better
than the piecewise constant isotonic estimator.

For the computational aspect, the proposed method consumes slightly more time
to complete the model estimation comparing to the SIC method. For instance,
the average elapsed time, over the 500 replications, on the model estimation
for experiment C with $\lambda_{C}=0.15$ and a sample size of $250$ ($500$)
are 12.28 (14.20) seconds for mSIC and 11.64 (13.67) seconds for SIC, with a
Core i7-1165G7 CPU laptop.

##  6 Real data application

We apply the proposed method to a dataset of melanoma patients extracted from
the Surveillance, Epidemiology and End Results (SEER) database. Such dataset
has also been studied by [23] and it consists of 1445 melanoma patients
diagnosed between 2004 – 2015. The event time of interest is the time to death
because of melanoma and the follow-up time ranges from 1 to 155 months. The
age of the patients varies between 11 to 104 years old. Among the 1445
patients, 596 are females and 849 are males. The cancer stage at diagnosis
(localized: 1302 cases, regional: 101 cases, distant: 42 cases) is also
recorded. The Kaplan-Meier estimate of the survival function in Figure 2 has a
long plateau, which contains around 20% of the observations. Combined with
medical evidence of possibility of cure for melanoma, this suggests that the
cure model is appropriate for this dataset.

Figure 2: Kaplan-Meier estimate of the survival function for the SEER data.

To compare the estimates of the logistic/Cox (LC), the single-index/Cox (SIC)
and the monotone single-index/Cox (mSIC) models, we split data into a training
set (with size of 964) and a testing set (with size of 481) and estimate the
models using the training set. The standardized age is considered in the
incidence in order to select a reasonable bandwidth for the SIC model. We note
that the SIC method selected an optimal bandwidth of $1$, which is the upper
bound of the search interval, $[0.4,1]$, considered in the SIC algorithm.
However, extending further the search interval for the bandwidth gave similar
results. Table 3 shows the parameter estimates, and the percentile-based
bootstrap confidence intervals with approximate 95% confidence level. The
confidence intervals are computed using 500 naive bootstrap samples. The
effects of the covariates have the same direction for all methods. Note that
the $\bm{\mathrm{\gamma}}$ estimate of the LC model are normalized, so that
the coefficients corresponding to the non-intercept terms possess a norm of
one for better comparison with the other two methods. To compare the
performance of the three models in predicting the uncure probability, we
compute the prediction error (PE) using the testing data

$PE=-\sum_{i=1}^{481}\hat{w}_{i}\log\left(1-\hat{p}(\bm{\mathrm{x}}_{i}^{\text{test}})\right)+(1-\hat{w}_{i})\log\hat{p}(\bm{\mathrm{x}}_{i}^{\text{test}}),$

where $\hat{w}_{i}$ is computed using (9) with the estimated parameters (and
the estimated link for SIC and mSIC.) The prediction errors are 95.47 for the
LC model, 101.14 for the SIC model and 81.46 for the mSIC model. This
indicates that mSIC performs better in predicting the uncure status among the
three approaches. As mentioned previously, the SIC algorithm selected an
optimal bandwidth of 1. The PE for SIC reduced slightly (to 97.15) when we
allowed the SIC algorithm to search for some larger bandwidth.

Table 3: Parameter estimates for the SEER data (LC’s $\hat{\bm{\mathrm{\gamma}}}_{n}$ is normalized) | Covariates | LC | SIC | mSIC
---|---|---|---|---
Est. |  Lower CI |  Upper CI | Est. |  Lower CI |  Upper CI | Est. |  Lower CI |  Upper CI
$\hat{\bm{\mathrm{\gamma}}}_{n}$ | Intercept | -4.2129 | -4.8391 | -3.3821 | - | - | 0.3640 | - | - | -
Age | 0.0620 | -0.0189 | 0.1463 | 0.1221 | 0.0586 | 0.3640 | 0.0487 | -0.0633 | 0.0843
Gender | 0.1298 | -0.0475 | 0.3019 | 0.2167 | -0.0132 | 0.7204 | 0.1713 | 0.0246 | 0.3119
Regional | 0.5842 | 0.4812 | 0.7208 | 0.5217 | 0.1992 | 0.6674 | 0.5658 | 0.4843 | 0.7121
Distant | 0.7987 | 0.7266 | 0.8935 | 0.8161 | 0.7393 | 1.1943 | 0.8051 | 0.7270 | 0.8794
$\hat{\bm{\mathrm{\beta}}}_{n}$ | Age | -0.0079 | -0.0277 | 0.0163 | -0.0086 | -0.0086 | -0.0283 | -0.0038 | -0.0196 | 0.0188
Gender | -0.1809 | -1.0741 | 0.9811 | -0.201 | -0.2010 | -1.0711 | -0.1609 | -0.8706 | 0.7711
Regional | 0.3908 | -0.5542 | 1.4638 | 0.479 | 0.4790 | -0.3423 | 0.2916 | -0.5600 | 1.0817
Distant | 1.4290 | 0.1771 | 2.5478 | 1.5064 | 1.5064 | 0.2783 | 1.1356 | -0.0604 | 2.0322
Figure 3: Link estimates (solid: LC, dotted: SIC, dashed: mSIC)

Figure 3 shows the link estimates of the three methods, plotted over the same
range of the index. The LC estimate is rescaled according to the normalization
of $\hat{\gamma}_{n}$. For this particular split of the data, all three
methods give a monotone link function, and the links of LC and mSIC have
similar shape. The estimated link function from SIC differs considerably from
the other two methods but it has the largest prediction error. However, as
illustrated in Appendix B.5, the SIC estimate is in general non-monotone and
unstable. Part of this behavior seems to be due to its sensitivity to the
bandwidth choice and the behavior of the cross-validation bandwidth selection
method. mSIC on the other hand, apart from guaranteeing a monotone link
estimate, is also more stable, less sensitive to the bandwidth and behaves
well even for a simple (non-optimal) bandwidth choice. To achieve a more
robust comparison among the three models using the prediction error, we also
generate 10 random splits of the data (training: 964, testing: 481) and
compute the prediction error (PE) using the testing data for each split. The
averages (standard deviations) of the prediction errors over the 10 random
splits are 89.64 (5.73) for LC; 91.84 (9.51) for SIC; and 71.84 (6.66) for
mSIC. This suggests the same conclusion as before that mSIC behaves better in
predicting the uncure probability among the three methods.

##  Appendix A Technical lemmas and proofs

###### Proof of Proposition 3.1.

We start by making an analogy with the standard isotonic regression problem,
which corresponds to solving the following minimization problem

$\hat{\bm{\mathrm{y}}}=\operatorname*{arg\,min}_{\bm{\mathrm{y}}\in\mathcal{C}}\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2},$

where $\mathcal{C}=\left\\{\bm{\mathrm{y}}\in[n]:\ y_{1}\leq
y_{2}\leq\cdots\leq y_{n}\right\\}$, for given observations
$x_{1},\dots,x_{n}$. From the theory of isotonic regression [27, Section 1.5],
it is known that the isotonic least squares estimate $\hat{\bm{\mathrm{y}}}$
is also solution of the following optimization problem

$\hat{\bm{\mathrm{y}}}=\operatorname*{arg\,min}_{\bm{\mathrm{y}}\in\mathcal{C}}\sum_{i=1}^{n}\Delta_{\Psi}\left(x_{i},y_{i}\right),$
(15)

where

$\Delta_{\Psi}\left(x,y\right)=\Psi\left(x\right)-\Psi\left(y\right)-\left(x-y\right)\psi\left(y\right),$
(16)

and $\Psi$ is a convex function with derivative $\psi$. This equivalence is
usually used to transform maximum likelihood isotonic estimation into an
isotonic least-squares problem. Moreover, $\hat{\bm{\mathrm{y}}}$ can be
characterized as the left derivative of the greatest convex minorant (GCM) of
the cumulative sum diagram (CSD)

$\left\\{\left(0,0\right),\left(i,\sum_{j=1}^{i}x_{(j)}\right),i=1,\cdots,n\right\\},$

where $x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}$ correspond to the ordered
observations. If in addition to the order restriction, a uniform bound
restriction is imposed on the solution, the problem becomes

$\hat{\bm{\mathrm{y}}}=\operatorname*{arg\,min}_{\bm{\mathrm{y}}\in\mathcal{C^{\prime}}}\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2},$

with $\mathcal{C}^{\prime}=\left\\{\bm{\mathrm{y}}\in[n]:\ a\leq y_{1}\leq
y_{2}\leq\cdots\leq y_{n}\leq b\right\\}$ and is studied by [15].
Specifically, it has shown that the minimizer under the order and uniform
bound restrictions is
$\tilde{\bm{\mathrm{y}}}=\left(\tilde{y}_{1},\cdots,\tilde{y}_{n}\right)$
given by

$\tilde{y}_{i}=\max\left(a,\min\left(\hat{y}_{i},b\right)\right),\qquad
i=1,\cdots,n,$

where $\hat{\bm{\mathrm{y}}}=\left(\hat{y}_{1},\cdots,\hat{y}_{n}\right)$ is
the solution of (15).

Going back to our maximization problem in (11), first note that since the
criterion depends only on the values of $\varphi$ at the observed points
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(1)}<\dots<\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(n)}$
we can only identify the correspondent values
$\bm{\mathrm{\phi}}=(\phi_{1},\dots,\phi_{n})$. If we choose the convex
function $\Psi\left(u\right)=u\log u+\left(1-u\right)\log\left(1-u\right)$ in
(16), we obtain

$\begin{split}&\sum_{i=1}^{n}\Delta_{\Psi}\left(w_{i},\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)\right)\\\
&=\sum_{i=1}^{n}\left\\{-w_{i}\log\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)-\left(1-w_{i}\right)\log\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i}\right)\right]\right\\}+C,\end{split}$
(17)

for some $C$ that depends on the observations and the fixed parameters but not
on $\varphi$. Therefore, the maximization problem in (11) is equivalent to
solving

$\widehat{\bm{\mathrm{\phi}}}=\operatorname*{arg\,min}_{\bm{\mathrm{\phi}}\in\mathcal{C}_{\epsilon^{\prime}}}\sum_{i=1}^{n}\Delta_{\Psi}\left(w_{(i)},\phi_{i}{}{}\right)$

where $\mathcal{C}_{\epsilon^{\prime}}=\left\\{\bm{\mathrm{\phi}}\in[n]:\
\epsilon^{\prime}\leq\phi_{1}\leq\phi_{2}\leq\cdots\leq\phi_{n}\leq
1-\epsilon^{\prime}\right\\}$. Consequently, using the previous results for
the isotonic regression problem, $\widehat{\bm{\mathrm{\phi}}}$ can be
characterized as

$\hat{\phi}_{i}=\max\left(\epsilon^{\prime},\min\left(\tilde{\phi}_{i},1-\epsilon^{\prime}\right)\right),\qquad
i=1,\cdots,n,$

where $\left(\tilde{\phi}_{1},\cdots,\tilde{\phi}_{n}\right)$ are the left
derivatives of the GCM of the CSD

$\left\\{\left(0,0\right),\left(i,\sum_{j=1}^{i}w_{(j)}\right),i=1,\cdots,n\right\\}$

with $w_{(j)}$ corresponding to the $j$-th order statistic of
$({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}_{i})_{i}$. Note also that the
procedure can easily accomodate ties in the observations of the index by
summing the correspondent $w_{j}$.

Finally we note that, if one would try to directly characterize the maximizer
of the likelihood criterion in (5) over the non-decreasing function $\varphi$
would still end up with an iterative procedure that is the same as the EM
algorithm. Indeed, the problem is equivalent to maximizing

$\mathcal{L}_{n}(\bm{\mathrm{\phi}})=\sum_{i=1}^{n}\Delta_{(i)}\log\phi_{i}+(1-\Delta_{(i)})\log\left\\{1-\phi_{i}+\phi_{i}S_{u}(Y_{(i)}|Z_{(i)})\right\\}$

over $\phi_{1}\leq\dots\leq\phi_{n}$ denoting again the values of the link
function on the ordered points
$\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(1)}<\dots<\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}}_{(n)}$.
If we assume to know

$W_{(i)}=\Delta_{(i)}+(1-\Delta_{(i)})\frac{\phi_{i}S_{u}(Y_{(i)}|Z_{(i)})}{1-\phi_{i}+\phi_{i}S_{u}(Y_{(i)}|Z_{(i)})}$

and choose $\Psi\left(u\right)=u\log u+\left(1-u\right)\log\left(1-u\right)$
as before, we obtain

$\sum_{i=1}^{n}\Delta_{\Psi}\left(W_{(i)},\phi_{i}\right)=-\mathcal{L}_{n}(\bm{\mathrm{\phi}})-\sum_{i=1}^{n}(1-\Delta_{(i)})W_{(i)}\log
S_{u}(Y_{(i)}|Z_{(i)}).$

As a result, maximizing $\mathcal{L}_{n}(\bm{\mathrm{\phi}})$ is equivalent to
minimizing the expression on the left hand side of the equation. The solution
can be characterized as the slope of the GCM of the cumulative sum diagram
$\\{(0,0),(i,\sum_{i=1}^{n}W_{(i)}),i=1,\cdots,n\\}$. However, the $W_{(i)}$
are actually not known and depend on $\bm{\mathrm{\phi}}$. One could construct
an iterative algorithm in which $\bm{\mathrm{\phi}}$ from the previous step is
used to compute $W_{(i)}$ and then update $\bm{\mathrm{\phi}}$ again. This is
the same as what EM algorithm does since the $W_{(i)}$ coincide with the ones
defined in (9). ∎

###### Deriavation of the characterization in (14).

By definition we have

$\begin{split}\varphi_{0,\bm{\mathrm{\theta}}}&=\operatorname*{arg\,max}_{\varphi\in\mathcal{M_{\epsilon^{\prime}}}}\mathbb{E}\left[\Delta\log\varphi(\bm{\mathrm{\gamma}}^{T}X)\right.\\\
&\left.\qquad\qquad\qquad+(1-\Delta)\log\left\\{1-\varphi(\bm{\mathrm{\gamma}}^{T}X)+\varphi(\bm{\mathrm{\gamma}}^{T}X)S_{u}(Y|Z;\bm{\mathrm{\beta}},\Lambda)\right\\}\right].\end{split}$

Let us assume for this that the covariates $X$ and $Z$ are independent and the
censoring is independent of the other variables. Define
$U=\bm{\mathrm{\gamma}}^{T}X$. Then the true likelihood for the observations
$(Y,\Delta,U,Z)$ is

$\begin{split}&\Delta\mathbb{P}\left(C>Y,B=1,T=Y\mid
U,Z\right)+(1-\Delta)\mathbb{P}\left(C=Y,T>Y\mid U,Z\right)\\\
&=\Delta\mathbb{P}\left(C>Y\right)\mathbb{P}\left(B=1\mid
U\right)\mathbb{P}\left(T=Y\mid B=1,Z\right)\\\
&\quad+(1-\Delta)\mathbb{P}\left(C=Y\right)\left\\{\mathbb{P}\left(B=0\mid
U\right)+\mathbb{P}\left(B=1\mid U\right)\mathbb{P}\left(T>Y\mid
B=1,Z\right)\right\\}\end{split}$

where with a slight abuse of notation we have used
$\mathbb{P}\left(\cdot\right)$ instead of the density function for the
continuous variables. By the law of conditional expectations we have

$\mathbb{P}\left(B=1\mid
U\right)=\mathbb{E}\left[\mathbb{E}\left[{\mathds{1}}\left\\{B=1\right\\}\mid
X\right]\mid
U\right]=\mathbb{E}\left[\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}X)\mid
U\right]=:\varphi^{*}_{\bm{\mathrm{\gamma}}}(U)$

Then the true log-likelihood for the observations $(Y,\Delta,U,Z)$ can be
written as

$\begin{split}&=\Delta\log\varphi^{*}_{\bm{\mathrm{\gamma}}}(U)+\Delta\log
f_{u}(Y|Z;\bm{\mathrm{\beta}}_{0},\Lambda_{0})\\\
&\quad+(1-\Delta)\log\left\\{1-\varphi^{*}_{\bm{\mathrm{\gamma}}}(U)+\varphi^{*}_{\bm{\mathrm{\gamma}}}(U)S_{u}(Y|Z;\bm{\mathrm{\beta}}_{0},\Lambda_{0})\right\\}+c\end{split}$

where $c$ denotes other terms that do not depend on the model parameters
(related to the censoring distribution). If we fix
$\bm{\mathrm{\theta}}=(\bm{\mathrm{\gamma}},\bm{\mathrm{\beta}}_{0},\Lambda_{0})$,
it follows from the Kullback-Leibler inequality that

$\operatorname*{arg\,max}_{\varphi\in\mathcal{M}}\mathbb{E}\left[\Delta\log\varphi(U)+(1-\Delta)\log\left\\{1-\varphi(U)+\varphi(U)S_{u}(Y|Z;\bm{\mathrm{\beta}}_{0},\Lambda_{0})\right\\}\right]$

is obtained for $\varphi=\varphi^{*}_{\bm{\mathrm{\gamma}}}$. On the other
hand, Assumptions (A4), (A1)(i-ii), (A3)(ii) imply that there exists
$\epsilon>0$ such that
$\epsilon\leq\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}X)\leq 1-\epsilon$. As a
result, if we choose $\epsilon^{\prime}<\epsilon$,
$\varphi^{*}_{\bm{\mathrm{\gamma}}}\in\mathcal{M_{\epsilon^{\prime}}}$. Hence
we conclude that under these more restrictive assumptions,
$\varphi_{0,\bm{\mathrm{\theta}}}$ is given by the expression in (14).

Next we comment on the necessity of such assumptions. To apply the previous
argument we needed that $C\perp(B,T)\mid U,Z$. Hence just conditional
independence $C\perp(B,T)\mid X,Z$ would not be sufficient to split the
probability into a product of probabilities. In addition, if $X$ and $Z$ would
be dependent then we also would not have $\mathbb{P}\left(B=1\mid
U,Z\right)=\mathbb{P}\left(B=1\mid U\right)$. For example, if $X=Z$, we would
have $\mathbb{P}\left(B=1\mid
U,Z\right)=\varphi_{0}(\bm{\mathrm{\gamma}}_{0}^{T}Z)$ and there would be no
connection between the function $\varphi^{*}_{\bm{\mathrm{\gamma}}}$ and the
true likelihood. Finally, if the latency parameters were not fixed to their
true values $\bm{\mathrm{\beta}}_{0},\Lambda_{0}$, then the Kullback-leibler
inequality could not be used to conclude that the maximizer is
$\varphi^{*}_{\bm{\mathrm{\gamma}}}$.

∎

###### Proof of Proposition 4.1.

Let $Q_{\bm{\mathrm{X}}}$ denote the distribution of $\bm{\mathrm{X}}$. For
any $\varphi\in\mathcal{M}_{\epsilon^{\prime}}$ and
$\bm{\mathrm{\gamma}}\in\Gamma$ define the $L_{2}$-norm

$\left\|{\varphi}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}}=\left(\int_{\mathcal{X}}\varphi(\bm{\mathrm{\gamma}}^{T}\bm{\mathrm{x}})^{2}\mathrm{d}{Q_{\bm{\mathrm{X}}}}\left(\bm{\mathrm{x}}\right)\right)^{\frac{1}{2}}$

To show the existence of a maximizer of
$l_{\bm{\mathrm{\theta}}}\left(\varphi\right)$ over
$\mathcal{M}_{\epsilon^{\prime}}$, it suffices to show that
$l_{\bm{\mathrm{\theta}}}$ is continuous on
$(\mathcal{M}_{\epsilon^{\prime}},\left\|{\cdot}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}})$
and
$(\mathcal{M}_{\epsilon^{\prime}},\left\|{\cdot}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}})$
is compact. To obtain uniqueness of the maximizer, we show that
$\varphi\longmapsto l_{\bm{\mathrm{\theta}}}\left(\varphi\right)$ is strictly
concave using the Gateaux derivative and $\mathcal{M}_{\epsilon^{\prime}}$ is
a convex set.

For simplicity we consider separately the two terms
$l_{1,\bm{\mathrm{\theta}}}\left(\varphi\right)=\mathbb{E}\left[\Delta\log\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]$
and
$l_{2,\bm{\mathrm{\theta}}}(\varphi)=\mathbb{E}[(1-\Delta)\log\\{1-\varphi({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}})F_{u}(Y|\bm{\mathrm{Z}})\\}]$.
Let $\left(\varphi_{m}\right)_{m}\subset\mathcal{M}_{\epsilon^{\prime}}$ be
any sequence on $\mathcal{M}_{\epsilon^{\prime}}$ that converges to
$\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$. For any $\eta>0$, we can
find an $N\left(\eta\right)\in\mathbb{N}$ such that
$\left\|{\varphi_{m}-\tilde{\varphi}}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}}<\eta$,
whenever $m\geq N\left(\eta\right)$. For $m\geq N\left(\eta\right)$, we have

$\displaystyle\begin{split}|l_{1,\bm{\mathrm{\theta}}}\left(\varphi_{m}\right)-l_{1,\bm{\mathrm{\theta}}}\left(\tilde{\varphi}\right)|&=\left|\mathbb{E}\left[\Delta\log\varphi_{m}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}})\right]-\mathbb{E}\left[\Delta\log\tilde{\varphi}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}})\right]\right|\\\
&\leq\frac{1}{\epsilon^{\prime}}\left(\mathbb{E}\left[\Delta^{2}\right]\mathbb{E}\left[\left\\{\varphi_{m}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\tilde{\varphi}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right\\}^{2}\right]\right)^{\frac{1}{2}}\\\
&\leq\frac{1}{\epsilon^{\prime}}\left\|{\varphi_{m}-\tilde{\varphi}}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}}<\frac{\eta}{\epsilon^{\prime}}.\end{split}$

Here, the second inequality follows from the Cauchy-Schwarz inequality, the
mean value theorem and the bound on
$\epsilon^{\prime}\leq\tilde{\varphi},\varphi_{m}\leq 1-\epsilon^{\prime}$. In
a similar way, for $m\geq N\left(\eta\right)$,

$\displaystyle\begin{split}|l_{2,\bm{\mathrm{\theta}}}\left(\varphi_{m}\right)-l_{2,\bm{\mathrm{\theta}}}\left(\tilde{\varphi}\right)|&\leq\frac{1}{\epsilon^{\prime}}\left(\mathbb{E}\left[\left\\{\left(1-\Delta\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right)\right\\}^{2}\right]\right.\\\
&\qquad\qquad\left.\times\mathbb{E}\left[\left\\{\varphi_{m}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\tilde{\varphi}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right\\}^{2}\right]\right)^{\frac{1}{2}}\\\
&\leq\frac{1}{\epsilon^{\prime}}\left\|{\varphi_{m}-\tilde{\varphi}}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}}<\frac{\eta}{\epsilon^{\prime}}.\end{split}$

Therefore, take $\xi={2\eta}/{\epsilon^{\prime}}>0$ and
$N=N\left({\xi\epsilon^{\prime}}/{2}\right)$, we have
$|l_{\bm{\mathrm{\theta}}}\left(\varphi_{m}\right)-l_{\bm{\mathrm{\theta}}}\left(\tilde{\varphi}\right)|<\xi$,
whenever $m\geq N$, and hence $l_{\bm{\mathrm{\theta}}}$ is continuous on
$(\mathcal{M}_{\epsilon^{\prime}},\left\|{\cdot}\right\|_{Q_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}})$.

Next, we show the compactness of
$(\mathcal{M}_{\epsilon^{\prime}},\left\|{\cdot}\right\|_{{Q}_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}})$.
By Helly’s selection theorem, since
$(\varphi_{m})_{m}\subset\mathcal{M}_{\epsilon^{\prime}}$ is a sequence of
uniformly bounded monotonically increasing functions, there exists a
convergent subsequence $(\varphi_{m_{k}})_{k\in\mathbb{N}}$ of
$(\varphi_{m})_{m\in\mathbb{N}}$ such that
$\varphi_{m_{k}}\left(u\right)\rightarrow\varphi^{\ast}\left(u\right)$ for
almost all $u\in$ with respect to the Lebesgue measure. $\varphi_{m_{k}}$ are
all uniformly bounded monotonically increasing functions, so as its limit
$\varphi^{\ast}$. By Assumption (A5), the distribution of
${\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}$ has a density with respect to the
Lebesgue measure, we have
$\left\|{\varphi_{m_{k}}-\varphi^{\ast}}\right\|_{Q_{\bm{\mathrm{X}}},\bm{\mathrm{\gamma}}}\rightarrow
0$ almost everywhere by the dominated convergence theorem.

We show the convexity of $\mathcal{M}_{\epsilon^{\prime}}$ by showing that the
convex combination $t\varphi_{1}+\left(1-t\right)\varphi_{2}$ belongs to
$\mathcal{M}_{\epsilon^{\prime}}$, for any
$\varphi_{1},\varphi_{2}\in\mathcal{M}_{\epsilon^{\prime}}$, and
$t\in\left[0,1\right]$. Let $-\infty<x\leq y<\infty$. We have
$\epsilon^{\prime}\leq\varphi_{1}\left(x\right)\leq\varphi_{1}\left(y\right)\leq
1-\epsilon^{\prime}$ and
$\epsilon^{\prime}\leq\varphi_{2}\left(x\right)\leq\varphi_{2}\left(y\right)\leq
1-\epsilon^{\prime}$. These inequalities imply that $\epsilon^{\prime}\leq
t\varphi_{1}\left(x\right)+\left(1-t\right)\varphi_{2}\left(x\right)\leq
t\varphi_{1}\left(y\right)+\left(1-t\right)\varphi_{2}\left(y\right)\leq
1-\epsilon^{\prime}$. Hence,
$t\varphi_{1}+\left(1-t\right)\varphi_{2}\in\mathcal{M}_{\epsilon^{\prime}}$.

Define
$H\left(t\right)=l_{\bm{\mathrm{\theta}}}\left(t\varphi_{1}+\left(1-t\right)\varphi_{2}\right)$,
where $0<t<1$ and $\varphi_{1},\varphi_{2}\in\mathcal{M}_{\epsilon^{\prime}}$
with $\varphi_{1}\neq\varphi_{2}$. We show that the second derivative of
$H\left(t\right)$ is negative on $\left(0,1\right)$ using the Gateaux
derivative and hence $\varphi\longmapsto
l_{\bm{\mathrm{\theta}}}\left(\varphi\right)$ is strictly concave. For
simplicity, denote $F_{u,0}=1-S_{u}\left(Y|\bm{\mathrm{Z}}\right)$ with the
true parameters $\bm{\mathrm{\beta}}_{0}$ and $\Lambda_{0}$, as in (3), and
$F_{u}=1-S_{u}\left(Y|\bm{\mathrm{Z}}\right)$ with parameters
$\bm{\mathrm{\beta}}$ and $\Lambda$. One can show that

$\begin{split}H^{\prime\prime}\left(t\right)&=\mathbb{E}\left[\frac{-\Delta\left[\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}}{\left[t\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)+\left(1-t\right)\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}}\right]\\\
&+\mathbb{E}\left[\frac{-\left(1-\Delta\right)F_{u}^{2}\left[\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}}{\left[1-\left\\{t\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)+\left(1-t\right)\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right\\}F_{u}\right]^{2}}\right].\end{split}$

Bounding the denominators by one and replacing $\Delta$ by

$\mathbb{E}\left[\Delta|\bm{\mathrm{X}},Y,\bm{\mathrm{Z}}\right]=\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\left(Y|\bm{\mathrm{Z}}\right)$

using the law of iterated expectations, we obtain

$H^{\prime\prime}\left(t\right)\leq-\mathbb{E}\left[\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}})F_{u,0}\left\\{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}})F_{u,0}\right\\}\left\\{\varphi_{1}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})-\varphi_{2}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})\right\\}^{2}\right].$
(18)

By Assumption (A3)(A3)(i), we have
$\inf_{\bm{\mathrm{x}}\in\mathcal{X}}\mathbb{P}\left[C>\tau_{0}|\bm{\mathrm{X}}=\bm{\mathrm{x}}\right]=c>0$.
Therefore, since
$\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{x}})\geq\epsilon$,

$\begin{split}\mathbb{E}\left[\Delta|\bm{\mathrm{X}}\right]&=\mathbb{P}\left[B=1,T\leq
C|\bm{\mathrm{X}}\right]\geq\mathbb{P}\left[B=1,C\geq\tau_{0}|\bm{\mathrm{X}}\right]=\mathbb{P}\left[B=1|\bm{\mathrm{X}}\right]\mathbb{P}\left[C\geq\tau_{0}|\bm{\mathrm{X}}\right]\\\
&\geq\epsilon\cdot\inf_{\bm{\mathrm{x}}\in\mathcal{X}}\mathbb{P}\left[C>\tau_{0}|\bm{\mathrm{X}}=\bm{\mathrm{x}}\right]=c\epsilon>0.\end{split}$
(19)

Thus, using the law of iterated expectations and
$\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\leq
1-\epsilon$, we obtain

$\displaystyle\begin{split}&\mathbb{E}\left[\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\left[1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\right]\left[\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}\right]\\\
&\geq\epsilon\cdot\mathbb{E}\left[\mathbb{E}\left[\Delta|\bm{\mathrm{X}},Y,\bm{\mathrm{Z}}\right]\left[\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}\right]\\\
&=\epsilon\cdot\mathbb{E}\left[\mathbb{E}\left[\Delta|\bm{\mathrm{X}}\right]\left[\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right]^{2}\right]\\\
&\geq
c\epsilon^{2}\cdot\mathbb{E}\left[\left\\{\varphi_{1}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)-\varphi_{2}\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right)\right\\}^{2}\right]>0.\end{split}$
(20)

Hence $H^{\prime\prime}\left(t\right)<0$, for $0<t<1$. ∎

###### Proof of Proposition 4.2.

If
$\bm{\mathrm{\theta}}=\bm{\mathrm{\theta}}_{0}=\left(\bm{\mathrm{\gamma}}_{0},\bm{\mathrm{\beta}}_{0},\Lambda_{0}\right)$,
by the definition of $\varphi_{0,\bm{\mathrm{\theta}}}$ in Proposition 4.1, we
have

$\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}}_{0},\varphi\right)\right]\leq\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}}_{0},\varphi_{0,\bm{\mathrm{\theta}}_{0}}\right)\right]\quad\text{for
all }\varphi\in\mathcal{M}_{\epsilon^{\prime}},$

with equality if $\varphi=\varphi_{0,\bm{\mathrm{\theta}}}$. From the
Kullback-Leibler inequality, we also have

$\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}}_{0},\varphi\right)\right]\leq\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}}_{0},\varphi_{0}\right)\right]\quad\text{for
all }\varphi\in\mathcal{M}_{\epsilon^{\prime}},$

with equality if $\varphi=\varphi_{0}$. Since, as shown in Proposition 4.1,
$\varphi_{0,\bm{\mathrm{\theta}}_{0}}$ is the unique maximizer of
$\varphi\mapsto\mathbb{E}\left[l\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}};\bm{\mathrm{\theta}}_{0},\varphi\right)\right]$,
it follows that $\varphi_{0,\bm{\mathrm{\theta}}_{0}}=\varphi_{0}$. ∎

###### Proof of Proposition 4.3.

We observe that, as in the standard logistic-Cox cure model, the maximizer
$\hat{\Lambda}_{n}$ must be a step function with jumps at the observed times.
Let $\lambda\left(t\right)$ be the jump size of $\Lambda$ at $t$. Also,
$u\longmapsto\hat{\varphi}_{n,\bm{\mathrm{\theta}}}^{s}\left(u\right)$ is
continuous on , therefore the likelihood function $L_{n}$ is continuous with
respect to $\bm{\mathrm{\gamma}}$, $\bm{\mathrm{\beta}}$ and the jump sizes of
$\Lambda$. Hence the existence and finiteness of the maximum likelihood
estimator
$(\hat{\bm{\mathrm{\gamma}}}_{n},\hat{\bm{\mathrm{\beta}}}_{n},\hat{\Lambda}_{n})$
follows as in the proof of Theorem 1 in [21]. ∎

###### Proof of Proposition 4.4.

Recall that
$\varphi_{0,\bm{\mathrm{\theta}}}=\operatorname*{arg\,max}_{\varphi\in\mathcal{M}_{\epsilon^{\prime}}}l_{\bm{\mathrm{\theta}}}\left(\varphi\right)$
for fixed $\bm{\mathrm{\theta}}$. Let
$\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$ and define
$H(t)=l_{\bm{\mathrm{\theta}}}\left(t\tilde{\varphi}+\left(1-t\right)\varphi_{0,\bm{\mathrm{\theta}}}\right)$,
where $0\leq t\leq 1$. By the definition of $\varphi_{0,\bm{\mathrm{\theta}}}$
and the convexity of $\mathcal{M}_{\epsilon^{\prime}}$, $H^{\prime}(0)\leq 0$
for all $\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$. We show the
continuity of $\varphi_{0,\bm{\mathrm{\theta}}}$ by contradiction.

For simplifying the notation, we denote
$\tilde{\varphi}=\tilde{\varphi}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}})$,
$\varphi_{0,\bm{\mathrm{\theta}}}=\varphi_{0,\bm{\mathrm{\theta}}}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}})$,
$F_{u,0}=F_{u}(Y|\bm{\mathrm{Z}};\bm{\mathrm{\beta}}_{0},\Lambda_{0})$, and
$F_{u}=F_{u}(Y|\bm{\mathrm{Z}};\bm{\mathrm{\beta}},\Lambda)$. Then, for all
$\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$,

$\displaystyle\begin{split}H^{\prime}(0)&=\mathbb{E}\left[\frac{\Delta\left(\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}\right)}{\varphi_{0,\bm{\mathrm{\theta}}}}-\frac{\left(1-\Delta\right)\left(\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}\right)F_{u}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}\right]\\\
&=\mathbb{E}\left[{\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}{\frac{\Delta-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}}\right]\\\
&=\mathbb{E}\left[{\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}{\frac{\mathbb{E}\left[\Delta|\bm{\mathrm{X}},Y,\bm{\mathrm{Z}}\right]-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}}\right]\\\
&=\mathbb{E}\left[{\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}{\frac{\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}}\right]\\\
&=\mathbb{E}\left[{\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}\left\\{1-\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right]\right\\}\right]\leq
0.\end{split}$ (21)

Next we argue that, if $\varphi_{0,\bm{\mathrm{\theta}}}$ was not continuous,
we can construct $\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$ for which
the inequality (21) is not satisfied. Hence, by contradiction we can conclude
that $\varphi_{0,\bm{\mathrm{\theta}}}$ is continuous.

Since $\varphi_{0,\bm{\mathrm{\theta}}}$ is monotone non-decreasing,
discontinuity points would be points of jump. Assume that
$\varphi_{0,\bm{\mathrm{\theta}}}$ has a jump at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$ with a size of $\kappa$.
Let

$A(u)=\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]$

We consider three cases based on whether
$A({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$ is smaller than one,
larger than one or equal to one.

Case 1. Suppose that $A({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})<1$.
By Assumption (A8), there exists $\eta>0$, such that $A(u)<1$ for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}+\eta})$.
Since $\varphi_{0,\bm{\mathrm{\theta}}}$ is monotone non-decreasing and
$\varphi_{0,\bm{\mathrm{\theta}}}$ has a jump at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$, we consider that
$\varphi_{0,\bm{\mathrm{\theta}}}$ is either right- or left-continuous at this
point and construct $\tilde{\varphi}$ which coincides with
$\varphi_{0,\bm{\mathrm{\theta}}}$ apart from in a small neighborhood of
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$.

  1. (i)

If $\varphi_{0,\bm{\mathrm{\theta}}}$ is right-continuous at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$,
$\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)<\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)$
for
$u\in[{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$.
We define
$\tilde{\varphi}\left(u\right)=\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)>\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$
for
$u\in[{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$
and
$\tilde{\varphi}\left(u\right)=\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$
otherwise.

  2. (ii)

If $\varphi_{0,\bm{\mathrm{\theta}}}$ is left-continuous at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$,
$\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)\leq\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)<\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa$
for
$u\in[{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$.
We define
$\tilde{\varphi}\left(u\right)=\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa>\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$
for
$u\in[{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$
and
$\tilde{\varphi}\left(u\right)=\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$
otherwise.

In addition, monotonicity of $\varphi_{0,\bm{\mathrm{\theta}}}$ implies that

$\begin{split}\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}(u)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]&\leq\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]=A(u)<1,\end{split}$

for
$u\in[{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$.
This inequality together with the construction of $\tilde{\varphi}$ in either
case give

$\begin{split}H^{\prime}\left(0\right)&=\mathbb{E}\left[{\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}\left\\{1-\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right]\right\\}\right]\\\
&=\mathbb{E}\left[{{\mathds{1}}\left\\{{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\frac{\eta}{2}\leq{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}<{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right\\}\frac{\tilde{\varphi}-\varphi_{0,\bm{\mathrm{\theta}}}}{\varphi_{0,\bm{\mathrm{\theta}}}}}\right.\\\
&\left.\qquad\qquad\left\\{1-\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}\right]\right\\}\right],\end{split}$

is strictly larger than zero, which contradicts the fact that
$H^{\prime}(0)\leq 0$ for all
$\tilde{\varphi}\in\mathcal{M}_{\epsilon^{\prime}}$.

Case 2. Suppose that $A({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})>1$.
By Assumption (A8), there exists $\eta>0$, such that $A(u)>1$ for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}+\eta})$.
Similar to the argument in the first case, we can show that

$\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}(u)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\geq\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]=A(u)>1$

for
$u\in({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}+\frac{\eta}{2}]$.
Similarly to the first case, we can construct $\tilde{\varphi}$ which
coincides with $\varphi_{0,\bm{\mathrm{\theta}}}$ apart from in a small
neighbourhood of ${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$ where it
is constant and strictly smaller than $\varphi_{0,\bm{\mathrm{\theta}}}$. We
then have $H^{\prime}\left(0\right)>0$ and leads to a contradiction.

Case 3. Suppose that $A({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})=1$.

(i) If $\varphi_{0,\bm{\mathrm{\theta}}}$ is right-continuous at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$, we have
$\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)\leq\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa<\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)$
for $u<{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$. This implies that
for $u<{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$,

$\displaystyle\begin{split}\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}(u)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]&\leq\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&<\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]=A(u).\\\
\end{split}$ (22)

We want to show that the left hand side of the previous inequality is smaller
than one in a left neighborhood of
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$ and then proceed with the
construction of $\tilde{\varphi}$ as in Case 1. By the mean value theorem, we
have

$\frac{1}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right)}-\frac{1}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}\left(Y|\bm{\mathrm{Z}}\right)}=\frac{\kappa
F_{u}\left(Y|\bm{\mathrm{Z}}\right)}{\left(1-\xi\right)^{2}}$

for some
$\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}\left(Y|\bm{\mathrm{Z}}\right)<\xi<\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right)$.
Since $\varphi_{0,\bm{\mathrm{\theta}}}\in\mathcal{M}_{\epsilon^{\prime}}$, we
have
$\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\geq\epsilon^{\prime}$
and $1>\xi>\epsilon^{\prime}F_{u}\left(Y|\bm{\mathrm{Z}}\right)\geq 0$. It
follows that

$\begin{split}&\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right)}-\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}\left(Y|\bm{\mathrm{Z}}\right)}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&\geq\mathbb{E}\left[\left.\left\\{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}\right\\}\kappa
F_{u}\left(Y|\bm{\mathrm{Z}}\right)\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&=\kappa\cdot\mathbb{E}\left[\left.\mathbb{E}\left[1-\Delta|\bm{\mathrm{X}},Y,\bm{\mathrm{Z}}\right]F_{u}\left(Y|\bm{\mathrm{Z}}\right)\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&\geq\kappa\cdot\mathbb{E}\left[\left.\left(1-\Delta\right)F_{u}\left(Y|\bm{\mathrm{Z}}\right){\mathds{1}}\left\\{Y>\tau_{0}\right\\}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&=\kappa\cdot\mathbb{P}\left[\left.Y>\tau_{0}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&=\kappa\cdot\mathbb{P}\left[\left.B=0,C>\tau_{0}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&=\kappa\cdot\mathbb{E}\left[\left.\mathbb{E}\left[\left.{\mathds{1}}\left\\{B=0\right\\}{\mathds{1}}\left\\{C>\tau_{0}\right\\}\right|\bm{\mathrm{X}}\right]\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&\geq c\epsilon\kappa>0.\end{split}$ (23)

Here the second equality follows from the definition of $\tau_{0}$ in
Assumption (A3)(A3)(i) and $F_{u}\left(y|\bm{\mathrm{z}}\right)=1$ if
$y>\tau_{0}$. The last inequality follows from
$\mathbb{P}\left[B=1,C\geq\tau_{0}|\bm{\mathrm{X}}\right]\geq c\epsilon$ as
shown in (19). By Assumption (A8), there exists $\eta>0$, such that
$A(u)<1+c\epsilon\kappa$ for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}+\eta})$.
Equation (23) implies that, for
$u\in({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,$
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})$, we have

$\displaystyle\begin{split}&\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]+c\epsilon\kappa\\\
&\leq\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]=A(u)<1+c\epsilon\kappa.\end{split}$

From (22), it follows that, for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}})$,

$\begin{split}&\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&\leq\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)-\kappa\right]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]<1.\end{split}$

By a similar construction of $\tilde{\varphi}$ as in the first case, we get
$H^{\prime}(0)>0$ which leads to a contradiction.

(ii) If $\varphi_{0,\bm{\mathrm{\theta}}}$ is left-continuous at
${\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$, we have
$\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)<\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa\leq\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)$
for $u>{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}$. This implies that

$\displaystyle\begin{split}\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]&\leq\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa\right]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\\\
&<\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right].\\\
\end{split}$

As in (i),

$\frac{1}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa\right]F_{u}\left(Y|\bm{\mathrm{Z}}\right)}-\frac{1}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}}=\frac{\kappa
F_{u}}{\left(1-\xi\right)^{2}}$

for some
$\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)F_{u}<\xi<\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa\right]F_{u}$.
Using a similar argument in (i), one can show that

$\mathbb{E}\left[\left.\frac{1-\varphi_{0}\left({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}}\right)F_{u,0}}{1-\left[\varphi_{0,\bm{\mathrm{\theta}}}\left({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}\right)+\kappa\right]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]>A(u)+c\epsilon\kappa.$

By Assumption (A8), there exists $\eta>0$, such that $A(u)>1-c\epsilon\kappa$
for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}-\eta,{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}+\eta})$.
Hence, we have for
$u\in({{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}},{\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}}}+\eta)$,

$\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-\varphi_{0,\bm{\mathrm{\theta}}}\left(u\right)F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]\geq\mathbb{E}\left[\left.\frac{1-\varphi_{0}({\bm{\mathrm{\gamma}}}^{T}_{0}\bm{\mathrm{X}})F_{u,0}}{1-[\varphi_{0,\bm{\mathrm{\theta}}}({\bm{\mathrm{\gamma}}}^{T}\tilde{\bm{\mathrm{x}}})+\kappa]F_{u}}\right|{\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{X}}=u\right]>1.$

By a similar construction of $\tilde{\varphi}$ as in the second case, we get
$H^{\prime}(0)>0$ and leads to a contradiction.∎

For the proof of Proposition 4.5 we need results on the entropy numbers of
certain classes of functions, which we derive below. First we introduce some
notation that will be used in the following series of Lemmas. Consider a class
of functions $\mathcal{F}$ equipped with a norm $\left\|{\cdot}\right\|$. For
$\zeta>0$, the bracketing number
$N_{B}\left({\zeta},{\mathcal{F}},{\left\|{\cdot}\right\|}\right)=N$ is the
minimal number of pairs of functions $\\{[{f_{j}^{L},f_{j}^{U}}]\colon
j=1,\cdots,N\\}$ such that $\|{f_{j}^{U}-f_{j}^{L}}\|\leq\zeta$, for all
$j=1,\cdots,N$ and for each $f\in\mathcal{F}$, there is a
$j\in\left\\{1,\cdots,N\right\\}$ such that $f_{j}^{L}\leq f\leq f_{j}^{U}$.
The $\zeta$–entropy with bracketing of $\mathcal{F}$ is the logarithm of the
bracketing number,
$H_{B}\left({\zeta},{\mathcal{F}},{\left\|{\cdot}\right\|}\right)=\log
N_{B}\left({\zeta},{\mathcal{F}},{\left\|{\cdot}\right\|}\right)$.
$\mathbb{P}_{0}$ denotes the distribution of
$\left(Y,\Delta,\bm{\mathrm{X}},\bm{\mathrm{Z}}\right)$. We will consider the
following classes of functions.

  * •

$\mathcal{M}$ is the class of all non-decreasing bounded functions on .

  * •

$\mathcal{F}$ is the class of functions
$F\left(y,\bm{\mathrm{z}}\right)=1-\exp\left[-\Lambda\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}\right)\right]$,
$y\in[0,\tau_{0}]$ and $\bm{\mathrm{z}}\in\mathcal{Z}$, where
$\Lambda\in\tilde{\mathcal{D}}$ and $\bm{\mathrm{\beta}}\in\mathcal{B}$.

  * •

$\mathcal{G}_{\epsilon^{\prime}}$ is the class of functions
$g\left(y,\delta,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)=\left(1-\delta\right)\log\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)F\left(y,\bm{\mathrm{z}}\right)\right]$,
$y\in[0,\tau_{0}]$, $\delta\in\left\\{0,1\right\\}$,
$\bm{\mathrm{x}}\in\mathcal{X}$ and $\bm{\mathrm{z}}\in\mathcal{Z}$, where
$\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}$,
$\varphi\in\mathcal{M}_{\epsilon^{\prime}}$ and $F\in\mathcal{F}$,

  * •

$\mathcal{H}_{\epsilon^{\prime}}$ is the class of functions
$h\left(\delta,\bm{\mathrm{x}}\right)=\delta\log\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)$,
$\delta\in\left\\{0,1\right\\}$ and $\bm{\mathrm{x}}\in\mathcal{X}$, where
$\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}$ and
$\varphi\in\mathcal{M}_{\epsilon^{\prime}}$.

  * •

$\mathcal{L}_{\epsilon^{\prime}}$ is the class of functions

$l\left(y,\delta,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)=\delta\log\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)+\left(1-\delta\right)\log\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)F\left(y,\bm{\mathrm{z}}\right)\right],$

$y\in[0,\tau_{0}]$, $\delta\in\left\\{0,1\right\\}$,
$\bm{\mathrm{x}}\in\mathcal{X}$ and $\bm{\mathrm{z}}\in\mathcal{Z}$, where
$\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}$,
$\varphi\in\mathcal{M}_{\epsilon^{\prime}}$ and $F\in\mathcal{F}$.

  * •

$\tilde{\mathcal{L}}_{\epsilon^{\prime}}$ is the class of functions

$\displaystyle\tilde{l}\left(y,\delta,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)$
$\displaystyle=\delta\left[\log\varphi_{1}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})-\log\varphi_{2}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})\right]$
$\displaystyle+(1-\delta)\\{\log[1-\varphi_{1}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})F(y,\bm{\mathrm{z}})]-\log[1-\varphi_{1}({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}})F(y,\bm{\mathrm{z}})]\\},$

for $y\in[0,\tau_{0}]$, $\delta\in\left\\{0,1\right\\}$,
$\bm{\mathrm{x}}\in\mathcal{X}$, $\bm{\mathrm{z}}\in\mathcal{Z}$,
$\bm{\mathrm{\gamma}}\in\mathcal{S}_{d-1}$,
$\varphi_{1},\varphi_{2}\in\mathcal{M}_{\epsilon^{\prime}}$ and
$F\in\mathcal{F}$.

We begin with a result from [31] and we use it to construct $\zeta$-brackets
for the other classes.

######  Lemma A.1 (Theorem 2.7.5 of [31] ).

There exists a constant $A>0$ such that

$H_{B}\left({\zeta},{\mathcal{M}},{\left\|{\cdot}\right\|_{r,Q}}\right)\leq\frac{A}{\zeta},$

for all $\zeta>0$, $r\geq 1$, and all probability measures $Q$ on , where
$\left\|{\cdot}\right\|_{r,Q}$ is the $L_{r}$–norm corresponding to $Q$.

######  Lemma A.2.

Suppose that Assumptions (A4) and (A10) are satisfied Let $\zeta>0$. There
exists a constant $C_{1}>0$ depending on $\bar{q}_{2}$, and $r_{2}$ only such
that

$H_{B}\left({\zeta},{\mathcal{F}},{\left\|{\cdot}\right\|_{1,\mathbb{P}_{0}}}\right)\leq\frac{C_{1}\left(q+1\right)}{\zeta}.$

Furthermore, there exists a constant $C_{2}>0$ depending on $\bar{q}_{1}$, and
$r_{1}$ and a constant $C_{3}>0$ depending on
$\epsilon^{\prime},\bar{q}_{1},\bar{q}_{2},r_{1}$, and $r_{2}$ such that

$H_{B}\left({\zeta},{\mathcal{G}_{\epsilon^{\prime}}},{\left\|{\cdot}\right\|_{1,\mathbb{P}_{0}}}\right)\leq\frac{C_{2}d+C_{3}\left(q+1\right)}{\zeta}.$

Moreover, there exists a constant $C_{4}>0$ depending on
$\epsilon^{\prime},\bar{q}_{1}$, and $r_{1}$ such that

$H_{B}\left({\zeta},{\mathcal{H}_{\epsilon^{\prime}}},{\left\|{\cdot}\right\|_{1,\mathbb{P}_{0}}}\right)\leq\frac{C_{4}\left(d+1\right)}{\zeta}.$

###### Proof.

Let $\zeta_{\bm{\mathrm{\beta}}}\in(0,1)$. Since $\mathcal{B}$ is a compact
subset of $[q]$, it can be covered by $N_{1}$ balls with radius
$\zeta_{\bm{\mathrm{\beta}}}$, where
$N_{1}\leq(A_{1}/\zeta_{\bm{\mathrm{\beta}}})^{q}$ for some constant
$A_{1}>0$. Let
$\left\\{\bm{\mathrm{\beta}}_{1},\cdots,\bm{\mathrm{\beta}}_{N_{1}}\right\\}$
be the centers of such balls. Consider
$f\left(y,\bm{\mathrm{z}}\right)=1-\exp\left[-\Lambda\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}\right)\right]\in\mathcal{F}$,
$y\in[0,\tau_{0}]$ and $\bm{\mathrm{z}}\in\mathcal{Z}$, for some
$\Lambda\in\tilde{\mathcal{D}}$ and $\bm{\mathrm{\beta}}\in\mathcal{B}$. We
can find $j\in\left\\{1,\cdots,N_{1}\right\\}$ such that
$\left\|{\bm{\mathrm{\beta}}-\bm{\mathrm{\beta}}_{j}}\right\|_{2}\leq\zeta_{\bm{\mathrm{\beta}}}$.
By the monotonicity of the exponential function and the Cauchy-Schwarz
inequality, we have

$\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\leq\exp\left({\bm{\mathrm{\beta}}}^{T}\bm{\mathrm{z}}\right)\leq\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right),$

for all $\bm{\mathrm{z}}\in\mathcal{Z}$. Let $\zeta_{\Lambda}>0$. By Lemma
A.1, the class $\tilde{\mathcal{D}}$ can be covered by
$\zeta_{\Lambda}$–brackets $[\Lambda_{l}^{L},\Lambda_{l}^{U}]$,
$l=1,\cdots,N_{2}$, such that

$\int_{[0,\tau_{0}]}\left[\Lambda_{l}^{U}\left(y\right)-\Lambda_{l}^{L}\left(y\right)\right]^{2}\mathrm{d}Q_{Y}\left(y\right)\leq\zeta_{\Lambda}^{2},$

where ${Q}_{Y}$ denotes the distribution of $Y$ and
$N_{2}\leq\exp{\left(A_{2}/\zeta_{\Lambda}\right)}$ for some constant
$A_{2}>0$. We note that $\Lambda_{l}^{L}$ and $\Lambda_{l}^{U}$ can always be
taken to be bounded below by 0 and bounded above by the uniform upper bound
$M$ of the class $\tilde{\mathcal{D}}$. Otherwise, we can take
$\Lambda_{l}^{L}\vee 0$ and $\Lambda_{l}^{U}\wedge M$. Then, we have

$1-\exp\left[-\Lambda_{l}^{L}(y)\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2})\right]\leq
f(y,\bm{\mathrm{z}})\leq
1-\exp\left[-\Lambda_{l}^{U}(y)\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2})\right],$

for some $l=1,\cdots,N_{2}$, and for all $y\in[0,\tau_{0}]$ and
$\bm{\mathrm{z}}\in\mathcal{Z}$. We then show that, for certain choices of
$\zeta_{\Lambda}$ and $\zeta_{\bm{\mathrm{\beta}}}$, the brackets

$\begin{split}[f_{l}^{L},f_{l}^{U}]=&\left[1-\exp\left\\{-\Lambda_{l}^{L}(y)\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2})\right\\},\right.\\\
&\quad
1-\left.\exp\left\\{-\Lambda_{l}^{U}(y)\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2})\right\\}\right]\end{split}$

are $\zeta$-brackets with respect to $\|\cdot\|_{1,\mathbb{P}_{0}}$ that cover
$\mathcal{F}$. We have

$\displaystyle\begin{split}\|f_{l}^{U}-f_{l}^{L}\|_{1,\mathbb{P}_{0}}=&\int_{[0,\tau_{0}]\times\mathcal{Z}}\bigg{|}\exp\left[-\Lambda_{l}^{U}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]\\\
&-\exp\left[-\Lambda_{l}^{L}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]\bigg{|}\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right).\end{split}$
(24)

Considering the integrand of (24), by the mean value theorem we obtain

$\displaystyle\begin{split}&\bigg{|}\exp\left[-\Lambda_{l}^{U}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]-\exp\left[-\Lambda_{l}^{L}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]\bigg{|}\\\
&=\bigg{|}\exp\left(-\xi\right)\left[\Lambda_{l}^{L}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)-\Lambda_{l}^{U}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]\bigg{|}\\\
&\leq\bigg{|}\Lambda_{l}^{L}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)-\Lambda_{l}^{U}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\bigg{|},\end{split}$

for some
$\xi\in(\Lambda_{l}^{L}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right),\Lambda_{l}^{U}\left(y\right)\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right))$.
As a result, from Minkowski inequality it follows that

$\displaystyle\begin{split}\|f_{l}^{U}-f_{l}^{L}\|_{1,\mathbb{P}_{0}}&\leq\int_{[0,\tau_{0}]\times\mathcal{Z}}\left|\Lambda_{l}^{L}\left(y\right)-\Lambda\left(y\right)\right|\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2})\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right)\\\
&+\int_{[0,\tau_{0}]\times\mathcal{Z}}\Lambda(y)\left|\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2})-\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2})\right|\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right)\\\
&+\int_{[0,\tau_{0}]\times\mathcal{Z}}\left|\Lambda\left(y\right)-\Lambda_{l}^{U}\left(y\right)\right|\exp({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2})\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right),\end{split}$
(25)

Consider the first integral on the right hand side of the last inequality in
(25). Using the Cauchy-Schwarz inequality, we have

$\displaystyle\begin{split}&\int_{[0,\tau_{0}]\times\mathcal{Z}}\left|\Lambda_{l}^{L}\left(y\right)-\Lambda\left(y\right)\right|\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right)\\\
&\leq\left\\{\int_{[0,\tau_{0}]}\left[\Lambda_{l}^{L}\left(y\right)-\Lambda\left(y\right)\right]^{2}\mathrm{d}Q_{Y}\left(y\right)\cdot\int_{\mathcal{Z}}\left[\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right]^{2}\mathrm{d}Q_{\bm{\mathrm{Z}}}\left(\bm{\mathrm{z}}\right)\right\\}^{\frac{1}{2}}\\\
&\leq\exp\left(b\right)\zeta_{\Lambda}\leq\exp\left(b+r_{2}\right)\zeta_{\Lambda},\end{split}$

where $b$ is such that the support of
${\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{Z}}$ is included in an interval
$\left[a,b\right]$ for all $j$. Similarly, we also have

$\int_{[0,\tau_{0}]\times\mathcal{Z}}\left|\Lambda\left(y\right)-\Lambda_{l}^{U}\left(y\right)\right|\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right)\leq\exp\left(b+r_{2}\right)\zeta_{\Lambda}.$

For the second integral on the right hand side of (25), we can show that

$\displaystyle\begin{split}&\int_{[0,\tau_{0}]\times\mathcal{Z}}\Lambda\left(y\right)\left|\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)-\exp\left({\bm{\mathrm{\beta}}}^{T}_{j}\bm{\mathrm{z}}+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\right|\mathrm{d}Q_{Y,\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{z}}\right)\\\
&\leq\bar{q}_{2}M\int_{a}^{b}\exp\left(t+\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)-\exp\left(t-\zeta_{\bm{\mathrm{\beta}}}r_{2}\right)\mathrm{d}t\\\
&=\bar{q}_{2}M\left(\int_{b-\zeta_{\bm{\mathrm{\beta}}}r_{2}}^{b+\zeta_{\bm{\mathrm{\beta}}}r_{2}}\exp\left(u\right)\mathrm{d}u-\int_{a-\zeta_{\bm{\mathrm{\beta}}}r_{2}}^{a+\zeta_{\bm{\mathrm{\beta}}}r_{2}}\exp\left(u\right)\mathrm{d}u\right)\\\
&\leq
2M\bar{q}_{2}\left[\exp\left(b+r_{2}\right)-\exp\left(a-r_{2}\right)\right]r_{2}\zeta_{\bm{\mathrm{\beta}}}\\\
&\leq
2M\bar{q}_{2}r_{2}\exp\left(b+r_{2}\right)\zeta_{\bm{\mathrm{\beta}}},\end{split}$

where $M$ denotes the upper bound of the class of functions
$\tilde{\mathcal{D}}$. If we take
$\zeta_{\Lambda}=\frac{\zeta}{3\exp\left(b+r_{2}\right)}$ and
$\zeta_{\bm{\mathrm{\beta}}}=\frac{\zeta}{6M\bar{q}_{2}r_{2}\exp(b+r_{2})}$ ,
we obtain

$\displaystyle\|f_{l}^{U}-f_{l}^{L}\|_{1,\mathbb{P}_{0}}\leq\zeta.$

Hence, using $\log x\leq x-1$, for $x>0$, we have

$\displaystyle\begin{split}H_{B}\left({\zeta},{\mathcal{F}},{\left\|{\cdot}\right\|_{1,\mathbb{P}_{0}}}\right)&\leq\log
N_{1}+\log N_{2}\\\ &\leq
q\log\left(\frac{6A_{1}M\bar{q}_{2}r_{2}\exp\left(b+r_{2}\right)}{\zeta}\right)+\frac{3A_{2}\exp\left(b+r_{2}\right)}{\zeta}\\\
&\leq\frac{C_{1}\left(q+1\right)}{\zeta}.\end{split}$

Next we construct brackets for the class of functions
$\mathcal{G}_{\epsilon^{\prime}}$ to show the second assertion of the Lemma.
Let $\zeta_{\bm{\mathrm{\gamma}}}>0$. By Lemma 7.5 of [3], $\mathcal{S}_{d-1}$
can be covered by $N_{3}$ neighborhoods with diameter at most
$\zeta_{\bm{\mathrm{\gamma}}}$, where
$N_{3}\leq\left(A_{3}/\zeta_{\bm{\mathrm{\gamma}}}\right)^{d}$ with a constant
$A_{3}>0$. Let
$\left\\{\bm{\mathrm{\gamma}}_{1},\cdots,\bm{\mathrm{\gamma}}_{N_{3}}\right\\}$
be elements of each these neighborhoods. Let $\zeta_{\varphi}>0$ and consider
$\zeta_{\varphi}$–brackets $[\varphi_{k}^{L},\varphi_{k}^{U}]$,
$k=1,\cdots,N_{4}$, covering the class $\mathcal{M}_{\epsilon^{\prime}}$ such
that

$\int\left|\varphi_{k}^{U}\left(t\right)-\varphi_{k}^{L}\left(t\right)\right|\mathrm{d}Q_{i}^{\pm}\left(t\right)\leq\zeta_{\varphi},\quad
k=1,\cdots,N_{4},$

where $Q_{i}^{\pm}\left(t\right)$ denotes the distribution of
${\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{X}}\pm\zeta_{\bm{\mathrm{\gamma}}}r_{1}$,
$i=1,\cdots,N_{3}$, and $N_{4}\leq\exp{\left(A_{4}/\zeta_{\varphi}\right)}$
for some constant $A_{4}>0$, by Lemma A.1. We note that $\varphi_{k}^{L}$ and
$\varphi_{k}^{U}$ can always be taken to be bounded below by
$\epsilon^{\prime}$ and bounded above by $1-\epsilon^{\prime}$, respectively.
Otherwise, we can take $\varphi_{k}^{L}\vee\epsilon^{\prime}$ and
$\varphi_{k}^{U}\wedge 1-\epsilon^{\prime}$. Using the monotonicity of
$\varphi$ and the Cauchy-Schwarz inequality, we have, for all
$\bm{\mathrm{x}}\in\mathcal{X}$,

$\varphi_{k}^{L}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)\leq\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)\leq\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}+\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right),$

for some $i=1,\cdots,N_{3}$ and $k=1,\cdots,N_{4}$. Using the result from the
first statement of the Lemma, we can consider a $\zeta_{F}$-bracket
$[F_{j}^{L},F_{j}^{U}]$, $j=1,\cdots,N$, covering the class $\mathcal{F}$ such
that

$\int\left|F_{j}^{U}\left(y|\bm{\mathrm{z}}\right)-F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right|\mathrm{d}\mathbb{P}_{0}\left(y,\delta,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)\leq\zeta_{F},\quad
j=1,\cdots,N,$

where $N\leq\exp[C_{1}(q+1)/\zeta_{F}]$. Then, we have

$\displaystyle\begin{split}\log\left[1-\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}+\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{U}\left(y|\bm{\mathrm{z}}\right)\right]&\leq\log\left[1-\varphi\left({\bm{\mathrm{\gamma}}}^{T}\bm{\mathrm{x}}\right)F_{u}\left(y|\bm{\mathrm{z}}\right)\right]\\\
&\leq\log\left[1-\varphi_{k}^{L}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right],\end{split}$

for some $j=1,\cdots,N$. We then show that, for a certain choice of
$\zeta_{F}$, the brackets

$\begin{split}[g_{i,j,k}^{L},g_{i,j,k}^{U}]&=\left[(1-\delta)\log\left\\{1-\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}+\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{U}\left(y|\bm{\mathrm{z}}\right)\right\\},\right.\\\
&\qquad\left.(1-\delta)\log\left\\{1-\varphi_{k}^{L}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right\\}\right]\end{split}$

are $\zeta$-brackets with respect to $\|\cdot\|_{1,\mathbb{P}_{0}}$ for the
class $\mathcal{G}_{\epsilon^{\prime}}$. Using the Minkowski inequality, we
have

$\begin{split}\|g_{i,j,k}^{U}-g_{i,j,k}^{L}\|_{1,\mathbb{P}_{0}}&\leq\int_{[0,\tau_{0}]\times\mathcal{X}\times\mathcal{Z}}\bigg{|}\log\left[1-\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right]\\\
&\qquad-\log\left[1-\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}+\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{U}\left(y|\bm{\mathrm{z}}\right)\right]\bigg{|}\mathrm{d}Q_{Y,\bm{\mathrm{X}},\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)\\\
&\quad+\int_{[0,\tau_{0}]\times\mathcal{X}\times\mathcal{Z}}\bigg{|}\log\left[1-\varphi_{k}^{L}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right]\\\
&\qquad-\log\left[1-\varphi_{k}^{U}\left({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1}\right)F_{j}^{L}\left(y|\bm{\mathrm{z}}\right)\right]\bigg{|}\mathrm{d}Q_{Y,\bm{\mathrm{X}},\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{x}},\bm{\mathrm{z}}\right).\\\
\end{split}$ (26)

For the second integral in (26) we have

$\displaystyle\int_{[0,\tau_{0}]\times\mathcal{X}\times\mathcal{Z}}\bigg{|}\log\left[1-\varphi_{k}^{L}({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1})F_{j}^{L}(y|\bm{\mathrm{z}})\right]$
$\displaystyle\qquad\qquad\qquad-\log\left[1-\varphi_{k}^{U}({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1})F_{j}^{L}(y|\bm{\mathrm{z}})\right]\bigg{|}\mathrm{d}Q_{Y,\bm{\mathrm{X}},\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)$
$\displaystyle\leq{\frac{1}{\epsilon^{\prime}}\int_{[0,\tau_{0}]\times\mathcal{X}\times\mathcal{Z}}\left|\varphi_{k}^{U}({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1})-\varphi_{k}^{L}({\bm{\mathrm{\gamma}}}^{T}_{i}\bm{\mathrm{x}}-\zeta_{\bm{\mathrm{\gamma}}}r_{1})\right|F_{j}^{L}(y|\bm{\mathrm{z}})\mathrm{d}Q_{Y,\bm{\mathrm{X}},\bm{\mathrm{Z}}}\left(y,\bm{\mathrm{x}},\bm{\mathrm{z}}\right)}$