# Extreme expectile estimation for short-tailed data, with an application to
market risk assessment

Abdelaati Daouiaa, Simone A. Padoanb & Gilles Stupflerc

(a Toulouse School of Economics, University of Toulouse Capitole, France
b Department of Decision Sciences, Bocconi University, via Roentgen 1, 20136
Milano, Italy
c Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France)

###### Abstract

The use of expectiles in risk management has recently gathered remarkable
momentum due to their excellent axiomatic and probabilistic properties. In
particular, the class of elicitable law-invariant coherent risk measures only
consists of expectiles. While the theory of expectile estimation at central
levels is substantial, tail estimation at extreme levels has so far only been
considered when the tail of the underlying distribution is heavy. This article
is the first work to handle the short-tailed setting where the loss (e.g.
negative log-returns) distribution of interest is bounded to the right and the
corresponding extreme value index is negative. We derive an asymptotic
expansion of tail expectiles in this challenging context under a general
second-order extreme value condition, which allows to come up with two
semiparametric estimators of extreme expectiles, and with their asymptotic
properties in a general model of strictly stationary but weakly dependent
observations. A simulation study and a real data analysis from a forecasting
perspective are performed to verify and compare the proposed competing
estimation procedures.

MSC 2010 subject classifications: 62G30, 62G32

Keywords: Expectiles, Extreme values, Second-order condition, Short tails,
Weak dependence

##  1 Introduction

The class of expectiles, introduced by Newey and Powell, (1987), defines
useful descriptors $\xi_{\tau}$ of the higher ($\tau\geq\frac{1}{2}$) and
lower ($\tau\leq\frac{1}{2}$) regions of the distribution of a random variable
$X$ through the asymmetric least squares (ALS) minimization problem

$\xi_{\tau}=\operatorname*{arg\,min}_{\theta\in\mathbb{R}}\mathbb{E}\big{[}\eta_{\tau}(X-\theta)-\eta_{\tau}(X)\big{]},$
(1.1)

where $\eta_{\tau}(x)=|\tau-\mathbbm{1}\\{x\leq 0\\}|\,x^{2}$, with
$\tau\in(0,1)$ and $\mathbbm{1}\\{\cdot\\}$ being the indicator function.
Expectiles are well-defined, finite and uniquely determined as soon the first
moment of $X$ is finite. They generalize the mean $\xi_{1/2}=\mathbb{E}(X)$ in
the same way quantiles generalize the median, thus defining an ALS analog to
quantiles. Indeed, Koenker and Bassett, (1978) showed that the $\tau$th
quantile $q_{\tau}$ of $X$ solves the asymmetric $L^{1}$ minimization problem

$q_{\tau}\in\operatorname*{arg\,min}_{\theta\in\mathbb{R}}\mathbb{E}(\varrho_{\tau}(X-\theta)-\varrho_{\tau}(X)),$

where $\varrho_{\tau}(x)=|\tau-\mathbbm{1}\\{x\leq 0\\}|\,|x|$. Expectiles
have received renewed attention for their ability to quantify tail risk at
least since the contribution of Taylor, (2008). They depend on the tail
realizations of $X$ and their probability, while quantiles only depend on the
frequency of tail realizations, see Kuan et al., (2009). Most importantly,
Ziegel, (2016) showed that expectiles are the sole coherent law-invariant
measure of risk which is also elicitable in the sense of Gneiting, (2011),
meaning that they abide by the intuitive diversification principle (Bellini et
al.,, 2014) and that their prediction can be performed through a
straightforward principled backtesting methodology. These merits have
motivated the development of procedures for expectile estimation and inference
over the last decade. A key, but difficult, question in any risk management
setup is the estimation of the expectile $\xi_{\tau}$ at extreme levels, which
grow to 1 as the sample size increases. This question was first tackled in
Daouia et al., (2018, 2020) under the assumption that the underlying
distribution is heavy-tailed, that is, its distribution function tends to 1
algebraically fast. The latest developments under this assumption have focused
on, among others, bias reduction (Girard et al.,, 2022), accurate inference
(Padoan and Stupfler,, 2022), and handling more complex data in regression
(Girard et al.,, 2021, 2022) or time series (Davison et al.,, 2022) setups.

The problem of estimating extreme expectiles outside of the set of heavy-
tailed models is substantially more complicated from a statistical standpoint.
The contribution of the present paper is precisely to build and analyze
semiparametric extreme expectile estimators in the challenging short-tailed
model, in which the extreme value index (EVI) of the underlying distributions
is known to be negative. This requires employing a dedicated extrapolation
relationship for population extreme expectiles. Only Mao et al., (2015) have
initiated such a study at the population level when $X$ belongs to the domain
of attraction of a Generalized Extreme Value distribution (GEV). Differently
to Mao et al., (2015), we work in the general semiparametric Generalized
Pareto (GP) setting through a standard second-order condition, which makes it
possible to derive an asymptotic expansion of extreme expectiles without
resorting to an unnecessary restriction about the link between the EVI and
second-order parameter that featured in Mao et al., (2015). Based on this
asymptotic expansion, we present and study two different extreme value
estimators of tail expectiles. The first one builds upon the Least
Asymmetrically Weighted Squares (LAWS) estimator of expectiles, namely the
empirical counterpart of $\xi_{\tau}$ in (1.1), obtained at intermediate
levels $\tau=\tau_{n}\to 1$ with $n(1-\tau_{n})\to\infty$ as the sample size
$n\to\infty$. The short-tail model assumption allows then to come up with an
expectile estimator extrapolated to the far tail at arbitrarily extreme levels
$\tau=1-p_{n}$ such that $(1-\tau_{n})/p_{n}\to\infty$ as $n\to\infty$, in a
semiparametric way reminiscent of how extreme quantiles are fitted in Section
4.3 of de Haan and Ferreira, (2006). The second extrapolating estimator
directly relies on the asymptotic expansion of $\xi_{\tau}$ that involves its
quantile analog $q_{\tau}$, the endpoint $q_{1}\equiv\xi_{1}$ and the EVI, by
plugging in the GP quantile-based estimators of these tail quantities. Our
estimation theory is valid in a general setting of strictly stationary and
weakly dependent data satisfying reasonable mixing and tail dependence
conditions. We explore various theoretical and practical features of extreme
expectile estimation in this setting, and explain why this problem is
statistically more difficult than extreme quantile estimation. In particular,
an extreme expectile $\xi_{\tau}$ is intrinsically less spread than its
quantile analog $q_{\tau}$, even at asymmetry levels $\tau\approx 1$ where it
remains much closer to the center of the distribution than $q_{\tau}$.
Consequently, any semiparametric procedure for extreme expectile estimation
should be expected to suffer at least from a worse bias than for extreme
quantile estimation.

Our focus on the problem of estimating extreme expectiles for bounded
distributions is motivated by the perhaps somewhat surprising finding that
weekly returns of equities, used in applications to circumvent the non-
synchronicity of daily data, may have short-tailed distributions.

Figure 1: Maximum Likelihood estimates of the extreme value index over the
resulting 253 successive rolling windows of $150$ stationary data, obtained
from 14 time series of weekly logarithmic (loss) returns between 21st
September 2014, and 12th June 2022.

This is illustrated in Figure 1 for 14 major companies and financial
institutions, where the data consists of the loss returns (i.e. negative log-
returns) on their weekly equity price from 21st September 2014 to 12th June
2022, corresponding to 403 trading weeks. The representative price is
constructed by averaging daily closing prices within the corresponding week.
The nature of the upper tail of these loss returns is reflected by the EVI of
their distribution whose negative, zero or positive values indicate
respectively a distribution with short, light or Pareto-type tail. None of
these three scenarios can be excluded in practice for these 14 data examples,
where the EVI is estimated on successive rolling windows of length $n=150$
using the GP distribution fitted to exceedances over a high threshold by means
of the Maximum Likelihood (ML) method, with the optimal threshold being chosen
by the path stability procedure as described below in Section 4. It is
therefore important to construct an appropriate and fully data-driven
estimation procedure for the challenging scenario of short-tailed data. This
problem also appears naturally in production econometrics when analyzing the
productivity of firms (Kokic et al.,, 1997). All our methods and data have
been incorporated into the R package ExtremeRisks.

In Section 2, we explain in detail the short tail distributional assumption on
$X$, state our asymptotic expansion linking extreme expectiles and quantiles,
construct our two classes of extreme expectile estimators and study their
asymptotic properties. A simulation study examines their finite-sample
performance in Section 3, and a time series of Bitcoin data is analyzed in
Section 4. The online Supplementary Material contains all the proofs in
Section A and further simulation results in Section B.

##  2 Main results

###  2.1 Connection between extreme expectiles and quantiles

Let $F:x\mapsto\mathbb{P}(X\leq x)$ be the distribution function of the random
variable of interest $X$ and $\overline{F}=1-F$ be its survival function.
Define the associated quantile function by
$q_{\tau}=\inf\\{x\in\mathbb{R}\,|\,F(x)\geq\tau\\}$ and the tail quantile
function $U$ by $U(s)=q_{1-s^{-1}}$, $s>1$. Differently from existing
literature on extreme expectile estimation, we focus on the case when the
distribution of $X$ is short-tailed, or equivalently, when its EVI $\gamma$ is
negative. According to Theorem 1.1.6 on p.10 of de Haan and Ferreira, (2006),
this corresponds to assuming that there is a positive function $a$ such that

$\forall z>0,\
\lim_{s\to\infty}\frac{U(sz)-U(s)}{a(s)}=\frac{z^{\gamma}-1}{\gamma},\mbox{
with }\gamma<0.$

This assumption can be informally rewritten as

$\forall z>0,\ U(sz)\approx U(s)+a(s)\frac{z^{\gamma}-1}{\gamma}\mbox{ when
}s\mbox{ is large.}$ (2.1)

This means that extreme values of $X$ at the far tail (represented by $U(sz)$)
can be achieved by extrapolating in-sample large values (represented by
$U(s)$) if the scale function $a(s)$ and the shape parameter $\gamma$ can be
consistently estimated. The theory of the resulting extreme value estimators
is usually developed under the following second-order refinement of the short-
tailed model assumption above, which will be our main condition throughout
(see de Haan and Ferreira,, 2006, Equation (2.3.13) on p.45): Condition
$\mathcal{C}_{2}(\gamma,a,\rho,A)\ $ There exist $\gamma<0$, $\rho\leq 0$, a
positive function $a(\cdot)$ and a measurable function $A(\cdot)$ having
constant sign and converging to $0$ at infinity such that, for all $z>0$,

$\lim_{s\to\infty}\frac{1}{A(s)}\left(\frac{U(sz)-U(s)}{a(s)}-\frac{z^{\gamma}-1}{\gamma}\right)=\int_{1}^{z}v^{\gamma-1}\left(\int_{1}^{v}u^{\rho-1}\mathrm{d}u\right)\mathrm{d}v.$

This condition enables one to control the bias incurred by using the
approximation (2.1) and represented by the function $A$. Under this condition,
the right endpoint $x^{\star}=\sup\\{x\in\mathbb{R}\,|\,F(x)<1\\}$ of $X$ is
necessarily finite (see de Haan and Ferreira,, 2006, Theorem 1.2.1 on p.19).
This justifies calling this model a short-tailed (or bounded) model.

Suppose now that $\mathbb{E}|\min(X,0)|<\infty$ and that condition
$\mathcal{C}_{2}(\gamma,a,\rho,A)$ is satisfied, so that
$\mathbb{E}|X|<\infty$ and expectiles of $X$ are well-defined and finite.
First, we motivate an asymptotic expansion of extreme expectiles that will be
instrumental in our subsequent theory of extreme expectile estimation. Recall
that the $\tau$th expectile $\xi_{\tau}$ satisfies

$\xi_{\tau}-\mathbb{E}(X)=\frac{2\tau-1}{1-\tau}\mathbb{E}((X-\xi_{\tau})\mathbbm{1}\\{X>\xi_{\tau}\\}),$
(2.2)

see Equation (12) in Bellini et al., (2014). Writing
$\mathbb{E}((X-x)\mathbbm{1}\\{X>x\\})$ as an integral of the quantiles of $X$
above $x$ and using condition $\mathcal{C}_{2}(\gamma,a,\rho,A)$ justifies the
approximation

$\mathbb{E}((X-\xi_{\tau})\mathbbm{1}\\{X>\xi_{\tau}\\})\approx\frac{\overline{F}(\xi_{\tau})a(1/\overline{F}(\xi_{\tau}))}{1-\gamma}\
\mbox{ as }\tau\uparrow 1,$

and therefore

$\lim_{\tau\uparrow
1}\frac{a(1/\overline{F}(\xi_{\tau}))\overline{F}(\xi_{\tau})}{1-\tau}=(x^{\star}-\mathbb{E}(X))(1-\gamma).$
(2.3)

The convergence $a(s)/(x^{\star}-U(s))\to-\gamma$ as $s\to\infty$ (see de Haan
and Ferreira,, 2006, Lemma 1.2.9 on p.22) then suggests

$\lim_{\tau\uparrow
1}\frac{(x^{\star}-\xi_{\tau})\overline{F}(\xi_{\tau})}{1-\tau}=(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1}).$
(2.4)

The approximations
$\overline{F}(\xi_{\tau})/(1-\tau)\approx\overline{F}(\xi_{\tau})/\overline{F}(q_{\tau})\approx(x^{\star}-\xi_{\tau})^{-1/\gamma}/(x^{\star}-q_{\tau})^{-1/\gamma}$
motivated by the regular variation property of
$x\mapsto\overline{F}(x^{\star}-1/x)$ (see de Haan and Ferreira,, 2006,
Theorem 1.2.1.2 on p.19) finally entail

$\lim_{\tau\uparrow
1}\frac{x^{\star}-\xi_{\tau}}{(x^{\star}-q_{\tau})^{1/(1-\gamma)}}=[(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})]^{-\gamma/(1-\gamma)}.$
(2.5)

Consequently, extreme expectiles can be extrapolated from their quantile
analogs in conjunction with endpoint and EVI estimation. Analyzing the
asymptotic properties of the estimators built in this way will require
quantifying the difference between the ratio
$(x^{\star}-\xi_{\tau})/(x^{\star}-q_{\tau})^{1/(1-\gamma)}$ and its limit in
(2.5). This is the focus of our first main result below.

######  Proposition 1.

Suppose that $\mathbb{E}|\min(X,0)|<\infty$ and condition
$\mathcal{C}_{2}(\gamma,a,\rho,A)$ holds with $\rho<0$, and let $x^{\star}$ be
the finite right endpoint of $F$. Then

$\displaystyle x^{\star}-\xi_{\tau}$
$\displaystyle=\frac{(x^{\star}-q_{\tau})^{\frac{1}{1-\gamma}}}{[(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})]^{\frac{\gamma}{1-\gamma}}}\Bigg{(}1-\left(\frac{x^{\star}-q_{\tau}}{(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})}\right)^{\frac{1}{1-\gamma}}(1+\operatorname{o}(1))$
$\displaystyle\qquad+\frac{\gamma[(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})]^{-\frac{\rho}{1-\gamma}}}{\rho(\gamma+\rho)(1-\gamma-\rho)}A\left(\frac{(x^{\star}-q_{\tau})^{\frac{1}{1-\gamma}}}{1-\tau}\right)(1+\operatorname{o}(1))\Bigg{)}$

as $\tau\uparrow 1$. In particular

$x^{\star}-\xi_{\tau}=\frac{(x^{\star}-q_{\tau})^{\frac{1}{1-\gamma}}}{[(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})]^{\frac{\gamma}{1-\gamma}}}\left(1+\operatorname{O}((1-\tau)^{-\frac{\gamma}{1-\gamma}})+\operatorname{O}(|A((1-\tau)^{-\frac{1}{1-\gamma}})|)\right).$

The additional condition $\rho<0$ in Proposition 1 is very mild and satisfied
in all standard short-tailed models, see Beirlant et al., (2004, Table 2.2 on
p.68).

######  Remark 1 (An equivalent asymptotic expansion).

An equivalent version of the asymptotic expansion in Proposition 1, where the
error terms rely solely on the expectile $\xi_{\tau}$, is

$x^{\star}-\xi_{\tau}=[(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})]^{-\gamma/(1-\gamma)}(x^{\star}-q_{\tau})^{1/(1-\gamma)}\\\
\times\left(1-\frac{x^{\star}-\xi_{\tau}}{(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})}(1+\operatorname{o}(1))+\frac{\gamma}{\rho(\gamma+\rho)(1-\gamma-\rho)}A(1/\overline{F}(\xi_{\tau}))(1+\operatorname{o}(1))\right).$

This is immediate from Proposition 1, (2.4) and (2.5), since $|A|$ is
regularly varying with index $\rho$.

######  Remark 2 (Comparison with Proposition 3.4 in Mao et al., (2015)).

Proposition 1 is an extension, tailored to our general semiparametric GP
setting and extended second-order regular variation assumption
$\mathcal{C}_{2}(\gamma,a,\rho,A)$, of Proposition 3.4 in Mao et al., (2015).
The latter result is formulated under a different, nonstandard second-order
regular variation condition on $\overline{F}$ when $X$ belongs to the domain
of attraction of a Generalized Extreme Value distribution. It is readily
checked by straightforward but tedious calculations that their quantities $c$,
$\gamma$, $\rho$ and $A(s)$ respectively correspond to (with the notation of
Lemma A.3 in Section A.1) $C^{1/\gamma}$, $-1/\gamma$, $-\rho/\gamma$ and
$-C^{-\rho/\gamma}A(s^{-1/\gamma})/(\gamma(\gamma+\rho))$ of the present
paper. In particular, when their asymptotic expansion applies, it coincides
with ours, but we lift an unnecessary restriction on the second-order
parameter $\rho$ that features in their result.

######  Remark 3 (Expectiles are less extreme than quantiles).

An immediate consequence of Equation (2.5) is that
$(x^{\star}-\xi_{\tau})/(x^{\star}-q_{\tau})\to\infty$ as $\tau\uparrow 1$,
that is, extreme quantiles are closer to the endpoint of a short-tailed
distribution than extreme expectiles. It is therefore unsurprising that the
bias due to the approximation of tail expectiles by their quantile analogs
under the second-order framework, which is asymptotically proportional to
$A((1-\tau)^{-1}(x^{\star}-q_{\tau})^{1/(1-\gamma)})$, converges more slowly
to 0 than the corresponding bias term in the heavy-tailed setting, whose order
is $A((1-\tau)^{-1})$, see Proposition 1(i) in Daouia et al., (2020). As a
second consequence, at least as far as handling bias is concerned, estimating
extreme expectiles under short-tailed models using a semiparametric extreme
value methodology should be expected to be much harder than under heavy-tailed
models.

######  Remark 4 (On remainder terms in the asymptotic expansion).

The quantity $x^{\star}-\mathbb{E}(X)$, which is a measure of the spread of
the distribution tail, appears in the asymptotic equivalent of
$(x^{\star}-\xi_{\tau})/(x^{\star}-q_{\tau})$ and in both the remainder terms
of the asymptotic expansion for $x^{\star}-\xi_{\tau}$. By contrast, no
measures of spread appear in the asymptotic connection between extreme
expectiles and quantiles of heavy-tailed distributions, although the
expectation $\mathbb{E}(X)$, which can be understood as a location parameter,
appears in an error term proportional to $1/q_{\tau}$, as can be seen from
Proposition 1 in Daouia et al., (2020).

With Proposition 1 at our disposal, we can now construct and study two classes
of extreme expectile estimators. The first one, in Section 2.2 below, is built
upon asymmetric least squares minimization, while the second one, in Section
2.3, is directly obtained by plugging in Equation (2.5) estimators of
$\mathbb{E}(X)$ and of the tail quantities $\gamma$, $x^{\star}$ and
$q_{\tau}$.

###  2.2 Asymmetric least squares estimation

Suppose that the available data has been generated from the random variables
$X_{1},\ldots,X_{n}$ with common distribution function $F$, and let
$\tau_{n}\uparrow 1$ (as $n\to\infty$) be a high asymmetry level at which the
target unknown expectile $\xi_{\tau_{n}}$ is to be estimated. A first solution
is to construct the estimator minimizing the empirical counterpart of problem
(1.1). This produces the Least Asymmetrically Weighted Squares (LAWS)
estimator

$\widehat{\xi}_{\tau_{n}}=\operatorname*{arg\,min}_{\theta\in\mathbb{R}}\frac{1}{n}\sum_{t=1}^{n}\eta_{\tau_{n}}(X_{t}-\theta)-\eta_{\tau_{n}}(X_{t})=\operatorname*{arg\,min}_{\theta\in\mathbb{R}}\sum_{t=1}^{n}\eta_{\tau_{n}}(X_{t}-\theta).$
(2.6)

A possible way to derive the asymptotic normality of
$\widehat{\xi}_{\tau_{n}}$ would be to find a quadratic approximation to
$\sum_{t=1}^{n}\eta_{\tau_{n}}(X_{t}-\theta)$ and then to use stochastic
convex optimization theorems such as those of Hjort and Pollard, (1993).
However, such developments already require lengthy technical arguments in the
better-known heavy-tailed model, as can be seen from Daouia et al., (2018).
Instead, we propose to use here an alternative technique leading to a much
simpler proof, which is based on the following observation made by Jones,
(1994): the $\tau$th expectile of $F$ is the $\tau$th quantile of the
distribution function $E=1-\overline{E}$, where

$\overline{E}(x)=\frac{\mathbb{E}(|X-x|\mathbbm{1}{\\{X>x\\}})}{\mathbb{E}(|X-x|)}.$

This survival function can equivalently be rewritten as

$\overline{E}(x)=\frac{\varphi^{(1)}(x)}{2\varphi^{(1)}(x)+x-\mathbb{E}(X)},\mbox{
with }\varphi^{(\kappa)}(x)=\mathbb{E}((X-x)^{\kappa}\mathbbm{1}{\\{X>x\\}}).$

Since $\widehat{\xi}_{\tau}$ is the $\tau$th expectile of the empirical
distribution function

$\widehat{F}_{n}(x)=\frac{1}{n}\sum_{t=1}^{n}\mathbbm{1}{\\{X_{t}>x\\}},$

it must therefore be the $\tau$th quantile of the distribution function
$\widehat{E}_{n}=1-\widehat{\overline{E}}_{n}$ defined as

$\widehat{\overline{E}}_{n}(x)=\frac{\widehat{\varphi}_{n}^{(1)}(x)}{2\widehat{\varphi}_{n}^{(1)}(x)+x-\overline{X}_{n}},\mbox{
where
}\widehat{\varphi}_{n}^{(\kappa)}(x)=\frac{1}{n}\sum_{t=1}^{n}(X_{t}-x)^{\kappa}\mathbbm{1}{\\{X_{t}>x\\}},$

with $\overline{X}_{n}$ being the sample mean. Intuitively, to derive now the
asymptotic behavior of $\widehat{\xi}_{\tau_{n}}-\xi_{\tau_{n}}$, it suffices
to obtain the asymptotic behavior of
$\widehat{\overline{E}}_{n}(x)/\overline{E}(x)$ at a level $x=x_{n}$ close to
$\xi_{\tau_{n}}$ in an appropriate sense and to apply a suitable inversion
argument.

We do so in a general framework of strictly stationary, weakly dependent
random variables. Recall that a strictly stationary sequence $(X_{t})_{t\geq
1}$ is said to be $\alpha-$mixing (or strongly mixing) if
$\alpha(l)=\sup_{m\geq 1}\alpha_{m}(l)\to 0$ when $l\to\infty$, where

$\alpha_{m}(l)=\sup_{\begin{subarray}{c}A\in\mathcal{F}_{1,m}\\\
B\in\mathcal{F}_{m+l,\infty}\end{subarray}}|\mathbb{P}(A\cap
B)-\mathbb{P}(A)\mathbb{P}(B)|$

with $\mathcal{F}_{1,m}=\sigma(X_{1},\ldots,X_{m})$ and
$\mathcal{F}_{m+l,\infty}=\sigma(X_{m+l},X_{m+l+1},\ldots)$ being past and
future $\sigma-$algebras. The $\alpha-$mixing condition is one of the weakest
dependence assumptions in the mixing time series literature: more restrictive
conditions include $\beta-$, $\rho-$, $\phi-$ and $\psi-$mixing, see Bradley,
(2005). We make the following assumption about the mixing rate. Condition
$\mathcal{M}\ $ There exist sequences of positive integers $(l_{n})$ and
$(r_{n})$, both tending to infinity, such that $l_{n}/r_{n}\to 0$, $r_{n}/n\to
0$ and $n\,\alpha(l_{n})/r_{n}\to 0$, as $n\to\infty$. The sequences $(l_{n})$
and $(r_{n})$ are respectively interpreted as “small-block” and “big-block”
sequences, and are used to develop a big-block/small-block argument as a
prerequisite to evaluating the asymptotic variance of
$\widehat{\xi}_{\tau_{n}}$. Condition $\mathcal{M}$ has already been used in
the literature on the extreme values of time series, see e.g. Rootzén et al.,
(1998, Equation (2.1)). We also require the following tail dependence
condition on the joint extreme behavior of $(X_{t})_{t\geq 1}$ at different
time points. Condition $\mathcal{D}\ $ For any integer $t\geq 1$, there is a
function $R_{t}$ on $[0,\infty]^{2}\setminus\\{(\infty,\infty)\\}$ such that

$\forall(x,y)\in[0,\infty]^{2}\setminus\\{(\infty,\infty)\\},\
\lim_{s\to\infty}s\,\mathbb{P}(\overline{F}(X_{1})\leq
x/s,\overline{F}(X_{t+1})\leq y/s)=R_{t}(x,y),$

and there exist a constant $K\geq 0$ and a nonnegative summable sequence
$(\rho(t))_{t\geq 1}$ such that, for $s$ large enough,

$\forall t\geq 1,\ \forall x,y\in(0,1],\ s\,\mathbb{P}(\overline{F}(X_{1})\leq
x/s,\overline{F}(X_{t+1})\leq y/s)\leq\rho(t)\sqrt{xy}+\frac{K}{s}xy.$

The function $R_{t}$, called the tail copula of $(X_{1},X_{t+1})$ (see Schmidt
and Stadtmüller,, 2006), finely quantifies the degree of asymptotic dependence
between $X_{1}$ and $X_{t+1}$. Condition $\mathcal{D}$ ensures that the
probability of a joint extreme value of $X_{1}$ and $X_{t+1}$ is of the same
order of magnitude as the probability of an extreme value of $X_{1}$, meaning
that clusters of extreme values across time cannot form too often. A similar
anti-clustering assumption is made in Drees, (2003), see conditions (C2) and
(C3) therein.

Under these temporal dependence assumptions and using our insight about the
link between the LAWS estimator and the empirical estimator of $\overline{E}$,
we can prove the following result on the joint asymptotic normality of the
LAWS estimator and an empirical quantile having the same order of magnitude.

######  Theorem 1.

Assume that $X$ satisfies condition $\mathcal{C}_{2}(\gamma,a,\rho,A)$. Let
$\tau_{n},\alpha_{n}\uparrow 1$ be such that
$n\overline{F}(\xi_{\tau_{n}})\to\infty$,
$\overline{F}(\xi_{\tau_{n}})/(1-\alpha_{n})\to 1$ and
$\sqrt{n\overline{F}(\xi_{\tau_{n}})}A(1/\overline{F}(\xi_{\tau_{n}}))=\operatorname{O}(1)$.

  1. (i)

Suppose that $(X_{t})_{t\geq 1}$ is a strictly stationary sequence of copies
of $X$, whose distribution function $F$ is continuous, satisfying conditions
$\mathcal{M}$ and $\mathcal{D}$. Assume that
$r_{n}\overline{F}(\xi_{\tau_{n}})\to 0$, and that there is $\delta>0$ such
that

$\mathbb{E}(|\min(X,0)|^{2+\delta})<\infty,\ \sum_{l\geq
1}l^{2/\delta}\alpha(l)<\infty\mbox{ and
}r_{n}\left(\frac{r_{n}}{\sqrt{n\overline{F}(\xi_{\tau_{n}})}}\right)^{\delta}\to
0.$

Then

$\frac{\sqrt{n\overline{F}(\xi_{\tau_{n}})}}{a(1/\overline{F}(\xi_{\tau_{n}}))}(\widehat{\xi}_{\tau_{n}}-\xi_{\tau_{n}},\widehat{q}_{\alpha_{n}}-q_{\alpha_{n}})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\bm{V}(\gamma)+2\,\bm{C}(\gamma,R))$

where the $2\times 2$ symmetric matrices $\bm{V}(\gamma)$ and
$\bm{C}(\gamma,R)$ are defined elementwise as
$V_{11}(\gamma)=2/[(1-\gamma)(1-2\gamma)]$, $V_{12}(\gamma)=1/(1-\gamma)$ and
$V_{22}(\gamma)=1$,

$\displaystyle C_{11}(\gamma,R)$
$\displaystyle=\frac{1}{\gamma^{2}}\iint_{(0,1]^{2}}\sum_{t=1}^{\infty}R_{t}(x^{-1/\gamma},y^{-1/\gamma})\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle C_{12}(\gamma,R)$
$\displaystyle=-\frac{1}{2\gamma}\int_{0}^{1}\sum_{t=1}^{\infty}[R_{t}(x^{-1/\gamma},1)+R_{t}(1,x^{-1/\gamma})]\,\mathrm{d}x$
$\displaystyle\mbox{and }C_{22}(\gamma,R)$
$\displaystyle=\sum_{t=1}^{\infty}R_{t}(1,1).$

  2. (ii)

If the $X_{i}$ are i.i.d. copies of $X$ and
$\mathbb{E}(|\min(X,0)|^{2})<\infty$, then the above asymptotic normality
result holds with $R_{t}\equiv 0$ for any $t\geq 1$, that is,

$\frac{\sqrt{n\overline{F}(\xi_{\tau_{n}})}}{a(1/\overline{F}(\xi_{\tau_{n}}))}(\widehat{\xi}_{\tau_{n}}-\xi_{\tau_{n}},\widehat{q}_{\alpha_{n}}-q_{\alpha_{n}})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\bm{V}(\gamma)).$

If $X$ is bounded, then assumption $\sum_{l\geq
1}l^{2/\delta}\alpha(l)<\infty$ in (i) can be weakened to $\sum_{l\geq
1}\alpha(l)<\infty$, and no integrability assumption on $X$ is necessary.

In Theorem 1, condition $n\overline{F}(\xi_{\tau_{n}})\to\infty$ requires that
$\tau_{n}$ be intermediate, i.e. not too large. Assumption
$\sqrt{n\overline{F}(\xi_{\tau_{n}})}A(1/\overline{F}(\xi_{\tau_{n}}))=\operatorname{O}(1)$
is a bias condition which corresponds exactly to the usual bias condition
$\sqrt{n(1-\tau_{n})}A((1-\tau_{n})^{-1})=\operatorname{O}(1)$ in extreme
quantile estimation when replacing $\xi_{\tau_{n}}$ by $q_{\tau_{n}}$, see
Theorem 2.4.1 on p.50 of de Haan and Ferreira, (2006). The conditions on
$r_{n}$ in Theorem 1(i) are similar to those of Theorem 3.1 in Davison et al.,
(2022) under heavy-tailed models, taking into account that
$\overline{F}(\xi_{\tau_{n}})$ is asymptotically proportional to $1-\tau_{n}$
in the latter setting. The integrability assumption on $X$ and the condition
on the mixing rate $\alpha(l)$ ensure that a central limit theorem applies to
$\overline{X}_{n}$, as part of the proof of the asymptotic normality of
$\widehat{\overline{E}}_{n}(x)/\overline{E}(x)$ at high levels $x=x_{n}$ close
to $\xi_{\tau_{n}}$.

######  Remark 5 (On bias conditions).

An inspection of the proof of Theorem 1(i) reveals that the bias condition
$\sqrt{n\overline{F}(\xi_{\tau_{n}})}A(1/\overline{F}(\xi_{\tau_{n}}))=\operatorname{O}(1)$
is only needed for the asymptotic normality of
$\widehat{q}_{\alpha_{n}}-q_{\alpha_{n}}$, and is unnecessary for the validity
of the asymptotic normality of $\widehat{\xi}_{\tau_{n}}-\xi_{\tau_{n}}$
alone.

######  Remark 6 (Comparison with the i.i.d. case).

It is natural to compare the asymptotic normality of the LAWS estimator for
short-tailed data with the corresponding result one obtains for heavy-tailed
data. We restrict the comparison to the i.i.d. setting for the sake of
simplicity. If $X$ has a heavy right tail, that is, $U(sz)/U(s)\to z^{\gamma}$
as $s\to\infty$, for any $z>0$, where $0<\gamma<1/2$, and under the
assumptions that $\mathbb{E}(|\min(X,0)|^{2+\delta})<\infty$ for some
$\delta>0$, $\tau_{n}\uparrow 1$ and $n(1-\tau_{n})\to\infty$, one has, by
Theorem 2 in Daouia et al., (2018),

$\sqrt{n(1-\tau_{n})}\left(\frac{\widehat{\xi}_{\tau_{n}}}{\xi_{\tau_{n}}}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,\frac{2\gamma^{3}}{1-2\gamma}\right).$

In this same setting, $(U(sz)-U(s))/a(s)\to(z^{\gamma}-1)/\gamma$ as
$s\to\infty$, with $a(s)=\gamma U(s)$, and
$\overline{F}(\xi_{\tau_{n}})/(1-\tau_{n})\to\gamma^{-1}-1=(1-\gamma)/\gamma$
(this was first shown by Bellini et al.,, 2014, Theorem 11). Therefore, when
$X$ has a heavy right tail,

$\frac{\sqrt{n\overline{F}(\xi_{\tau_{n}})}}{a(1/\overline{F}(\xi_{\tau_{n}}))}(\widehat{\xi}_{\tau_{n}}-\xi_{\tau_{n}})\approx\frac{\sqrt{1-\gamma}}{\gamma^{3/2}}\times\sqrt{n(1-\tau_{n})}\left(\frac{\widehat{\xi}_{\tau_{n}}}{\xi_{\tau_{n}}}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,\frac{2(1-\gamma)}{1-2\gamma}\right).$

It follows that the rates of convergence of the LAWS estimator look similar in
both heavy and bounded tail settings, but there is a phase transition in terms
of asymptotic variance: the term $1-\gamma$ appears in its numerator for heavy
tails, while it appears in the denominator for short tails, as established in
Theorem 1. Interestingly, the two asymptotic variances in the heavy and short-
tailed settings converge to 2, and therefore exactly match together in the
light-tailed middle scenario, when $\gamma\to 0$.

######  Remark 7 (Comparison with the asymptotic normality of intermediate
quantiles).

It is instructive to compare Theorem 1 with the asymptotic normality result
for the direct intermediate quantile estimator at level $\tau_{n}$, namely,
the empirical counterpart $\widehat{q}_{\tau_{n}}=X_{\lceil
n\tau_{n}\rceil,n}$. According to Theorem 2.4.1 on p.50 of de Haan and
Ferreira, (2006), when the $X_{i}$ are i.i.d.,

$\frac{\sqrt{n\overline{F}(q_{\tau_{n}})}}{a(1/\overline{F}(q_{\tau_{n}}))}(\widehat{q}_{\tau_{n}}-q_{\tau_{n}})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,1).$

Observe that, by a combination of Lemma 1.2.9 in de Haan and Ferreira, (2006)
and Lemma A.1 in Section A.1, $a(1/\overline{F}(x))/(x^{\star}-x)\to-\gamma$
as $x\uparrow x^{\star}$, and therefore

$\left.\frac{\sqrt{n\overline{F}(\xi_{\tau_{n}})}}{a(1/\overline{F}(\xi_{\tau_{n}}))}\right/\frac{\sqrt{n\overline{F}(q_{\tau_{n}})}}{a(1/\overline{F}(q_{\tau_{n}}))}=\sqrt{\frac{\overline{F}(\xi_{\tau_{n}})}{\overline{F}(q_{\tau_{n}})}}\times\frac{x^{\star}-q_{\tau_{n}}}{x^{\star}-\xi_{\tau_{n}}}(1+\operatorname{o}(1)).$

By (2.4) and (2.5) this ratio is asymptotically proportional to
$(x^{\star}-q_{\tau_{n}})^{-(\gamma+1/2)/(1-\gamma)}$ under the mild further
condition $\rho<0$. In other words, the intermediate LAWS estimator
$\widehat{\xi}_{\tau_{n}}$ converges faster than $\widehat{q}_{\tau_{n}}$ when
$\gamma>-1/2$, has the same rate of convergence if $\gamma=-1/2$, and
converges at a slower rate if $\gamma<-1/2$.

As a corollary of Theorem 1, we obtain the asymptotic normality of the
empirical estimator $\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}})$ of
$\overline{F}(\xi_{\tau_{n}})$, on which the rate of convergence of
$\widehat{\xi}_{\tau_{n}}$ crucially depends.

######  Corollary 1.

Work under the conditions of Theorem 1. Then

$\sqrt{n\overline{F}(\xi_{\tau_{n}})}\left(\frac{\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}})}{\overline{F}(\xi_{\tau_{n}})}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,\frac{2\gamma^{2}+\gamma+1}{(1-\gamma)(1-2\gamma)}\right.\\\
\left.+2\iint_{(0,1]^{2}}\sum_{t=1}^{\infty}\left(\frac{1}{\gamma^{2}}R_{t}(x^{-1/\gamma},y^{-1/\gamma})+\frac{1}{\gamma}[R_{t}(x^{-1/\gamma},1)+R_{t}(1,x^{-1/\gamma})]+R_{t}(1,1)\right)\mathrm{d}x\,\mathrm{d}y\right).$

######  Remark 8 (On rates of convergence).

The rate of convergence of
$\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}})$ is rather natural: for
a sequence $(u_{n})$ tending to $x^{\star}$ such that
$n\overline{F}(u_{n})\to\infty$, Lemma A.5 states that

$\sqrt{n\overline{F}(u_{n})}\left(\frac{\widehat{\overline{F}}_{n}(u_{n})}{\overline{F}(u_{n})}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}\left(0,1+2\sum_{t=1}^{\infty}R_{t}(1,1)\right).$

It is worth noticing that the asymptotic variance of
$\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}})$ does not coincide with
the variance that would be obtained if $\xi_{\tau_{n}}$ were known, namely, if
$\widehat{\overline{F}}_{n}(\xi_{\tau_{n}})$ were considered instead. This is
due to the asymptotic dependence existing between $\widehat{\xi}_{\tau_{n}}$
and high order statistics (and therefore between $\widehat{\xi}_{\tau_{n}}$
and $\widehat{\overline{F}}_{n}$), see Theorem 1 and the proof of Corollary 1.

We now have the tools necessary to construct an extreme value estimator of a
properly extreme expectile $\xi_{1-p_{n}}$, where $p_{n}\downarrow 0$ at any
possible rate as $n\to\infty$, by extrapolating the intermediate LAWS
estimator $\widehat{\xi}_{\tau_{n}}$ to the right place at the far tail. Using
(2.1) with $s=1/\overline{F}(\xi_{\tau_{n}})$ and
$z=\overline{F}(\xi_{\tau_{n}})/\overline{F}(\xi_{1-p_{n}})$ motivates the
approximation

$\xi_{1-p_{n}}\approx\xi_{\tau_{n}}+a(1/\overline{F}(\xi_{\tau_{n}}))\frac{(\overline{F}(\xi_{\tau_{n}})/\overline{F}(\xi_{1-p_{n}}))^{\gamma}-1}{\gamma}.$

By Theorem 1, $\xi_{\tau_{n}}$ is estimated by the LAWS estimator
$\widehat{\xi}_{\tau_{n}}$ at rate
$a(1/\overline{F}(\xi_{\tau_{n}}))/\sqrt{n\overline{F}(\xi_{\tau_{n}})}$. The
scale parameter $a(1/\overline{F}(\xi_{\tau_{n}}))$ and shape parameter
$\gamma$ can be estimated by a variety of techniques, such as the GP-pseudo-ML
estimator and Moment-type estimator, see Sections 3.4 and 3.5 in de Haan and
Ferreira, (2006). Typical such estimators of the scale function
$a(1/\overline{F}(u_{n}))$, when $u_{n}\uparrow x^{\star}$ such that
$n\overline{F}(u_{n})\to\infty$, converge on the relative scale at the rate
$1/\sqrt{n\overline{F}(u_{n})}$; see Sections 3.4 and 4.2 in de Haan and
Ferreira, (2006) in the i.i.d. case, and Section 6 in Drees, (2003) in the
dependent data setup. Since, by Corollary 1, the (unknown) quantity
$\overline{F}(u_{n})=\overline{F}(\xi_{\tau_{n}})$ can be consistently
estimated at the rate $1/\sqrt{n\overline{F}(\xi_{\tau_{n}})}$, we therefore
expect to be able to estimate $a(1/\overline{F}(\xi_{\tau_{n}}))$ at this rate
on the relative scale. Finally, given an intermediate level $\tau_{n}$, it is
customary to estimate the extreme value index $\gamma$ at the rate
$1/\sqrt{n(1-\tau_{n})}$ when the top $k=\lfloor n(1-\tau_{n})\rfloor$ values
in the data are used, see Sections 3.4, 3.5 and 3.6 in de Haan and Ferreira,
(2006) in the i.i.d. case, and again Section 6 in Drees, (2003) when the data
points are serially dependent. It remains to find a way to estimate
$\overline{F}(\xi_{\tau_{n}})/\overline{F}(\xi_{1-p_{n}})$, which depends on
the target quantity $\xi_{1-p_{n}}$ itself. Combining (2.4) and (2.5) with the
fact that the function $s\mapsto x^{\star}-U(s)$ is regularly varying with
index $\gamma$ (de Haan and Ferreira,, 2006, Corollary 1.2.10 on p.23)
suggests that

$\displaystyle\frac{\overline{F}(\xi_{\tau_{n}})}{\overline{F}(\xi_{1-p_{n}})}\approx\frac{1-\tau_{n}}{p_{n}}\times\frac{x^{\star}-\xi_{1-p_{n}}}{x^{\star}-\xi_{\tau_{n}}}$
$\displaystyle\approx\frac{1-\tau_{n}}{p_{n}}\left(\frac{x^{\star}-q_{1-p_{n}}}{x^{\star}-q_{\tau_{n}}}\right)^{1/(1-\gamma)}$
$\displaystyle\approx\frac{1-\tau_{n}}{p_{n}}\left(\frac{1-\tau_{n}}{p_{n}}\right)^{\gamma/(1-\gamma)}=\left(\frac{1-\tau_{n}}{p_{n}}\right)^{1/(1-\gamma)}$
(2.7)

which in turn leads to the expectile-specific approximation

$\xi_{1-p_{n}}\approx\xi_{\tau_{n}}+a(1/\overline{F}(\xi_{\tau_{n}}))\frac{((1-\tau_{n})/p_{n})^{\gamma/(1-\gamma)}-1}{\gamma}.$

Consequently, like extreme quantiles, extreme expectiles can be extrapolated
from their values at lower levels. More specifically, given estimators
$\widehat{\sigma}_{n}$ and $\widehat{\gamma}_{n}$ of
$a(1/\overline{F}(\xi_{\tau_{n}}))$ and $\gamma$, respectively, one can then
construct the $\xi_{1-p_{n}}$ estimator

$\widehat{\xi}_{1-p_{n}}^{\star}=\widehat{\xi}_{\tau_{n}}+\widehat{\sigma}_{n}\frac{((1-\tau_{n})/p_{n})^{\widehat{\gamma}_{n}/(1-\widehat{\gamma}_{n})}-1}{\widehat{\gamma}_{n}}.$
(2.8)

Since $(1-\tau_{n})/\overline{F}(\xi_{\tau_{n}})\to 0$, the parameter $\gamma$
is estimated at a slower rate than the other quantities, so we expect the
asymptotic behavior of $\widehat{\gamma}_{n}$ to govern that of
$\widehat{\xi}_{1-p_{n}}^{\star}$. The last theorem of this section makes this
intuition rigorous. Its proof crucially relies on Theorem 1 and on Proposition
1 in order to quantify the bias in the approximation (2.7).

######  Theorem 2.

Work under the conditions of Theorem 1. If moreover $\rho<0$,
$n(1-\tau_{n})\to\infty$, $(1-\tau_{n})/p_{n}\to\infty$,
$\sqrt{n(1-\tau_{n})}/\log((1-\tau_{n})/p_{n})\to\infty$,
$\sqrt{n(1-\tau_{n})}(x^{\star}-q_{\tau_{n}})^{1/(1-\gamma)}=\operatorname{O}(1)$,
$\sqrt{n(1-\tau_{n})}A((1-\tau_{n})^{-1})=\operatorname{O}(1)$,
$\widehat{\sigma}_{n}$ and $\widehat{\gamma}_{n}$ are such that

$\sqrt{n\overline{F}(\xi_{\tau_{n}})}\left(\frac{\widehat{\sigma}_{n}}{a(1/\overline{F}(\xi_{\tau_{n}}))}-1\right)=\operatorname{O}_{\mathbb{P}}(1)\
\mbox{ and }\
\sqrt{n(1-\tau_{n})}(\widehat{\gamma}_{n}-\gamma)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\Gamma,$

where $\Gamma$ is a nondegenerate limit, then

$\frac{\sqrt{n(1-\tau_{n})}}{a(1/\overline{F}(\xi_{\tau_{n}}))}(\widehat{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\frac{\Gamma}{\gamma^{2}}.$

###  2.3 Quantile-based estimation

We use here Proposition 1 to present an alternative estimator of extreme
expectiles, purely based on quantiles, and to develop its asymptotic theory.
Similarly to the setup of extreme quantile estimation in Section 4.3 of de
Haan and Ferreira, (2006), assume that $k=k_{n}\to\infty$ is a sequence of
positive integers such that $k/n\to 0$ and that respective estimators
$\widehat{\gamma}_{n}$, $\widehat{a}(n/k)$ and $X_{n-k,n}$ of $\gamma$,
$a(n/k)$ and $U(n/k)$ are given such that

$\sqrt{k}\left(\widehat{\gamma}_{n}-\gamma,\frac{\widehat{a}(n/k)}{a(n/k)}-1,\frac{X_{n-k,n}-U(n/k)}{a(n/k)}\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}(\Gamma,\Lambda,B)$ (2.9)

where $(\Gamma,\Lambda,B)$ is a nontrivial trivariate weak limit. This
assumption is satisfied by moment and GP-pseudo-ML estimators of the shape and
scale parameters, among others, see an overview in Section 4.2 of de Haan and
Ferreira, (2006) in the case where the $X_{i}$ are independent random
variables. It is also satisfied when $(X_{t})_{t\geq 1}$ is a strictly
stationary but serially dependent sequence: this is for example the case when
the data points are $\beta-$mixing and satisfy an anti-clustering condition
similar to the tail dependence assumption $\mathcal{D}$, as a consequence of
the powerful results of Drees, (2003).

Let $p_{n}\downarrow 0$ with $k/(np_{n})\to\infty$, so that the level
$1-p_{n}$ is much more extreme than $1-k/n$. Following Section 4.3 in de Haan
and Ferreira, (2006), the extreme quantile $q_{1-p_{n}}$ and the right
endpoint $x^{\star}$ can be estimated by

$\widehat{q}_{1-p_{n}}^{\star}=X_{n-k,n}+\widehat{a}(n/k)\frac{(k/(np_{n}))^{\widehat{\gamma}_{n}}-1}{\widehat{\gamma}_{n}}\
\mbox{ and }\
\widehat{x}^{\star}=X_{n-k,n}-\frac{\widehat{a}(n/k)}{\widehat{\gamma}_{n}}.$
(2.10)

According to Proposition 1, an estimator of $\xi_{1-p_{n}}$ follows then as

$\widetilde{\xi}_{1-p_{n}}^{\star}=\widehat{x}^{\star}-[(\widehat{x}^{\star}-\overline{X}_{n})(1-\widehat{\gamma}_{n}^{-1})]^{-\widehat{\gamma}_{n}/(1-\widehat{\gamma}_{n})}(\widehat{x}^{\star}-\widehat{q}_{1-p_{n}}^{\star})^{1/(1-\widehat{\gamma}_{n})}.$
(2.11)

The next result provides its asymptotic properties, where two sequences
$(u_{n})$ and $(v_{n})$ are said to be asymptotically proportional if
$(u_{n}/v_{n})$ tends to a finite positive limit as $n\to\infty$.

######  Theorem 3.

Suppose that $\mathbb{E}|\min(X,0)|<\infty$ and condition
$\mathcal{C}_{2}(\gamma,a,\rho,A)$ holds with $\rho<0$, and let $x^{\star}$ be
the finite right endpoint of $F$. Assume that condition (2.9) holds true and
that
$\sqrt{k}(\overline{X}_{n}-\mathbb{E}(X))\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$
with $k=k_{n}$ being asymptotically proportional to $n^{\alpha}$, for some
$\alpha\in(0,1)$. Let $p_{n}$ be asymptotically proportional to $n^{-\beta}$
where $\beta>0$ is such that $\alpha+\beta-1>0$. If moreover
$\sqrt{k}A(n/k)\to\lambda\in\mathbb{R}$, then we have, up to changing
probability spaces and with appropriate versions of the estimators involved,

$\displaystyle\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}$
$\displaystyle=\frac{a(\frac{n}{k})}{\sqrt{k}}\frac{1}{\gamma^{2}}\left(\Gamma+\gamma^{2}B-\gamma\Lambda-\frac{\lambda\gamma}{\gamma+\rho}+\operatorname{o}_{\mathbb{P}}(1)\right)$
$\displaystyle-\left[a\left(\frac{n}{k}\right)\left(\frac{k}{np_{n}}\right)^{\gamma}\right]^{\frac{1}{1-\gamma}}\frac{(1-\gamma)^{-\frac{1}{1-\gamma}}}{\gamma(x^{\star}-\mathbb{E}(X))^{\frac{\gamma}{1-\gamma}}}\left(\frac{\log(\frac{np_{n}^{1/(1-\gamma)}}{k})}{\sqrt{k}}\Gamma+\operatorname{o}_{\mathbb{P}}\left(\frac{\log
n}{\sqrt{k}}\right)\right)$
$\displaystyle+\operatorname{O}(n^{\beta\gamma/(1-\gamma)}(n^{\beta\gamma/(1-\gamma)}+|A(n^{\beta/(1-\gamma)})|)).$

######  Remark 9 (Mixing and the central limit theorem).

Condition
$\sqrt{k}(\overline{X}_{n}-\mathbb{E}(X))\stackrel{{\scriptstyle\mathbb{P}}}{{\longrightarrow}}0$
is satisfied in practice if
$\sqrt{n}(\overline{X}_{n}-\mathbb{E}(X))=\operatorname{O}_{\mathbb{P}}(1)$,
which is in particular true when a central limit theorem applies. As already
highlighted below Theorem 1, this will be the case if
$\mathbb{E}(|\min(X,0)|^{2})<\infty$ when the $X_{i}$ are independent, or if
there is $\delta>0$ such that $\mathbb{E}(|\min(X,0)|^{2+\delta})<\infty$ and
$\sum_{l\geq 1}l^{2/\delta}\alpha(l)<\infty$ when $(X_{t})_{t\geq 1}$ is
$\alpha-$mixing. In particular, when the data mixes geometrically fast, then
$\sqrt{n}(\overline{X}_{n}-\mathbb{E}(X))=\operatorname{O}_{\mathbb{P}}(1)$ as
soon as $X$ has a finite moment of order $2+\delta$, for some $\delta>0$.

######  Remark 10 (Our assumptions on $k$).

The assumption that $k=k_{n}$ is asymptotically equivalent to a positive and
finite multiple of $n^{\alpha}$, is only very slightly stronger than the usual
pair of extreme value conditions $k\to\infty$ and $k/n\to 0$ made throughout
Section 4 in de Haan and Ferreira, (2006). The only difference is that our
assumption does not allow to take $k$ growing to infinity logarithmically
fast; such sequences produce, however, very small values of $k$ in practice
and would therefore yield estimators having very large variances. We also note
that in standard settings such as those of Beirlant et al., (2004, Table 2.2
on p.68), $A(s)$ is asymptotically proportional to $s^{\rho}$, in which case
the optimal choices of $k$ satisfying the usual bias-variance tradeoff for
extreme value index estimation would fulfill
$\sqrt{k}A(n/k)\to\lambda\in\mathbb{R}\setminus\\{0\\}$, that is,
$k=\operatorname{O}(n^{-2\rho/(1-2\rho)})$. In other words, it is reasonable
to expect that optimal choices of $k$ in practice have to be asymptotically
equivalent to a positive and finite multiple of a fractional power of $n$.

It follows from Theorem 3 that the asymptotic behavior of the extreme
expectile estimator $\widetilde{\xi}_{1-p_{n}}^{\star}$ is more complex than
that of the extreme quantile estimator $\widehat{q}_{1-p_{n}}^{\star}$: while,
from Theorem 4.3.1 on p.134 and Theorem 4.5.1 on p.146 of de Haan and
Ferreira, (2006), $\widehat{q}_{1-p_{n}}^{\star}-q_{1-p_{n}}$ converges to the
same distribution
$\frac{1}{\gamma^{2}}\left(\Gamma+\gamma^{2}B-\gamma\Lambda-\lambda\frac{\gamma}{\gamma+\rho}\right)$
as $\widehat{x}^{\star}-x^{\star}$ at the rate $a(n/k)/\sqrt{k}$ for
$\gamma<0$, the asymptotic distribution of
$\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}$ may be a nonstandard mixture
of the two limiting distributions of $\widehat{x}^{\star}-x^{\star}$ and
$\widehat{\gamma}_{n}-\gamma$. In particular, Corollary 2 shows that when, for
example, $\beta=1$ and $\alpha$ (and hence $k$) is chosen small enough, it is
in fact the asymptotic distribution $\Gamma$ of $\widehat{\gamma}_{n}-\gamma$
that dominates in $\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}$, while
Corollary 3 examines what can otherwise be said.

######  Corollary 2.

Work under Theorem 3. If $\alpha<1-\beta/(1-\gamma)$ and
$\alpha<2\beta\min(-\gamma,-\rho)/(1-\gamma)$,

$\frac{\sqrt{k}}{\log(np_{n}^{1/(1-\gamma)}/k)}\frac{\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}}{[a(\frac{n}{k})(\frac{k}{np_{n}})^{\gamma}]^{1/(1-\gamma)}}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}-\gamma^{-1}(1-\gamma)^{-1/(1-\gamma)}(x^{\star}-\mathbb{E}(X))^{-\gamma/(1-\gamma)}\Gamma.$

######  Remark 11 (Link between bias terms and short-tailedness).

The closer $\gamma$ is to 0, the stronger the constraint on $\alpha$ through
the condition $\alpha<2\beta\min(-\gamma,-\rho)/(1-\gamma)$. This is analogous
to what happens in extreme expectile estimation for heavy-tailed
distributions, where the condition $\sqrt{k}/q_{1-k/n}=\operatorname{O}(1)$
(see e.g. Daouia et al.,, 2020, Theorem 5) becomes a strong restriction as the
tail gets less heavy, i.e. when $\gamma$ approaches 0.

######  Remark 12 (Comparison between the LAWS and quantile-based estimators).

One may compare the rates of convergence of $\widehat{\xi}_{1-p_{n}}^{\star}$
and $\widetilde{\xi}_{1-p_{n}}^{\star}$ by setting $\tau_{n}=1-k/n$. Using the
convergence $a(s)/(x^{\star}-U(s))\to-\gamma$ as $s\to\infty$ and Equation
(2.5), one finds under the assumptions of Corollary 2 that

$\frac{\sqrt{k}}{a(1/\overline{F}(\xi_{1-k/n}))}\left/\frac{\sqrt{k}}{\log(np_{n}^{1/(1-\gamma)}/k)[a(n/k)(k/(np_{n}))^{\gamma}]^{1/(1-\gamma)}}\right.\\\
\propto\log(n)\frac{[a(n/k)]^{1/(1-\gamma)}}{a(1/\overline{F}(\xi_{1-k/n}))}(k/(np_{n}))^{\gamma/(1-\gamma)}\propto
n^{(\alpha+\beta-1)\gamma/(1-\gamma)}\log(n)\to 0.$

This means that $\widetilde{\xi}_{1-p_{n}}^{\star}$ converges to
$\xi_{1-p_{n}}$ faster than $\widehat{\xi}_{1-p_{n}}^{\star}$ when $k$ (or
$1-\tau_{n}$) is chosen small. We shall illustrate this finding below in our
simulation study.

Condition $\alpha<1-\beta/(1-\gamma)$ may not hold in a given example,
especially when $\beta$ is large enough, or equivalently, $p_{n}$ is small
enough. Yet, interestingly this condition can always be satisfied for
sufficiently small $\alpha$ in the standard setting $\beta=1$ of extreme value
analysis. If it is not satisfied, then $\widetilde{\xi}_{1-p_{n}}^{\star}$
tends to inherit the asymptotic behavior of $\widehat{x}^{\star}$, rather than
$\widehat{\gamma}$, as established in the following result.

######  Corollary 3.

Under the assumptions of Theorem 3, if moreover $\alpha>1-\beta/(1-\gamma)$,
then

$\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}=\frac{a(\frac{n}{k})}{\sqrt{k}\gamma^{2}}\left(\Gamma+\gamma^{2}B-\gamma\Lambda-\frac{\lambda\gamma}{\gamma+\rho}+\operatorname{o}_{\mathbb{P}}(1)\right)+\operatorname{O}(n^{\frac{\beta\gamma}{1-\gamma}}(n^{\frac{\beta\gamma}{1-\gamma}}+|A(n^{\frac{\beta}{1-\gamma}})|)).$

The condition $\alpha>1-\beta/(1-\gamma)$ itself is not sufficient to ensure
the convergence of $\widetilde{\xi}_{1-p_{n}}^{\star}$; in practice, the bias
term may dominate the asymptotics depending on the choice of $k$. This is most
easily seen when $A(s)$ is asymptotically proportional to $s^{\rho}$ and
$\beta=1$, corresponding to the standard extreme value situation where
$p_{n}\approx c/n$. In this case, (i) one automatically has
$\alpha+\beta-1=\alpha>0$, (ii) condition
$\sqrt{k}A(n/k)\to\lambda\in\mathbb{R}$ essentially amounts to
$\alpha\leq-2\rho/(1-2\rho)$, and (iii) condition $\alpha>1-\beta/(1-\gamma)$
becomes $\alpha>-\gamma/(1-\gamma)$. For the bias term in Corollary 3 to be
negligible, one requires

$\frac{\sqrt{k}}{a(n/k)}\times
n^{\gamma/(1-\gamma)}(n^{\gamma/(1-\gamma)}+|A(n^{1/(1-\gamma)})|)\to 0.$

Since $a(n/k)$ is asymptotically proportional to $(n/k)^{\gamma}$ by Lemma
A.3(i), this is equivalent to assuming

$\alpha\left(\frac{1}{2}+\gamma\right)+\frac{\gamma^{2}-\min(-\gamma,-\rho)}{1-\gamma}<0.$

When $\gamma>-1/2$, as is often the case in applications, and
$0<-\rho<-\gamma$, representing situations where the bias due to the second-
order framework is high, this condition becomes

$\alpha<-\frac{2(\gamma^{2}+\rho)}{(1-\gamma)(1+2\gamma)}.$

Depending on the value of $\rho$, this final condition may not be compatible
with $\alpha>-\gamma/(1-\gamma)$: in fact, if $\rho$ is close enough to 0, it
may even be impossible to satisfy whatever the value of $\alpha$ (since the
right-hand side of the above displayed inequality tends to a negative constant
as $\rho\to 0$, when $\gamma>-1/2$). In this case, with the choice
$p_{n}=c/n$, the asymptotic behavior of $\widehat{x}^{\star}-x^{\star}$ can
never dominate in $\widetilde{\xi}_{1-p_{n}}^{\star}-\xi_{1-p_{n}}$.

###  2.4 Selection of the expectile asymmetry level

In practical situations it is crucial to make an informed decision as to what
the asymmetry level of the target expectile should be. In financial
applications, where the dual interpretation of expectiles in terms of the
gain-loss ratio is available (Bellini and Di Bernardino,, 2017), it is
sensible to set the expectile level so as to achieve a certain value of the
gain-loss ratio. Otherwise, it has been proposed in the literature to select
$\tau$ such that $\xi_{\tau}$ coincides with another pre-specified intuitive
risk measure: Bellini and Di Bernardino, (2017) suggest to choose the
expectile level $\tau$ so that $\xi_{\tau}$ is identical to the Value-at-Risk
(or quantile) $q_{\alpha}$, where $\alpha$ is a high tail probability level
specified by the statistician or the practitioner.

The proposal of Bellini and Di Bernardino, (2017) is valid only when the
underlying loss distribution is Gaussian. Daouia et al., (2018) later extended
this idea to the heavy-tailed setup. We examine here the short-tailed
situation, hitherto unexplored from this perspective. Fix a large quantile
level $\alpha=\alpha_{n}=1-p_{n}$. Setting $\tau=\tau_{n}=1-\pi_{n}$ to be the
expectile level such that $\xi_{\tau}=q_{\alpha}$, Equation (2.4) leads to

$\frac{(x^{\star}-q_{1-p_{n}})p_{n}}{\pi_{n}}=\frac{(x^{\star}-\xi_{\tau})\overline{F}(\xi_{\tau})}{1-\tau}\approx(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1}).$

In other words,

$\pi_{n}\approx\frac{x^{\star}-q_{1-p_{n}}}{(x^{\star}-\mathbb{E}(X))(1-\gamma^{-1})}p_{n}.$

This approximation suggests to estimate the quantity $\pi_{n}$ by

$\widehat{\pi}_{n}\equiv\widehat{\pi}_{n}(p_{n})=\frac{\widehat{x}^{\star}-\widehat{q}_{1-p_{n}}^{\star}}{(\widehat{x}^{\star}-\overline{X}_{n})(1-\widehat{\gamma}_{n}^{-1})}p_{n}$

with the notation of (2.10). Our final main result shows that this estimator
is asymptotically normal in the framework of Section 2.3.

######  Proposition 2.

Work under Theorem 3. If moreover
$\alpha<\min(-2\beta\gamma,-2\rho/(1-2\rho))$, then

$\frac{\sqrt{k}}{\log(k/(np_{n}))}\left(\frac{\widehat{\pi}_{n}}{\pi_{n}}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\Gamma.$

######  Remark 13 (Comparison with the heavy-tailed setting).

In the heavy-tailed case, according to Section 5 in Daouia et al., (2018),
$\pi_{n}\approx p_{n}/(\gamma^{-1}-1)$. An estimator of $\pi_{n}$ is then
$\widehat{\pi}_{n}=p_{n}/(\widehat{\gamma}_{n}^{-1}-1)$. In this setting, it
is straightforward to obtain, under a suitable bias condition when
$\sqrt{k}(\widehat{\gamma}_{n}-\gamma)\to\Gamma$, that

$\sqrt{k}\left(\frac{\widehat{\pi}_{n}}{\pi_{n}}-1\right)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\frac{\Gamma}{\gamma(1-\gamma)}.$

The estimator $\widehat{\pi}_{n}$ therefore converges at a slightly faster
rate in the heavy-tailed model. The slower speed of convergence in the short-
tailed framework is due to the presence of the quantity
$\widehat{x}^{\star}-\widehat{q}_{1-p_{n}}^{\star}=-\widehat{a}(n/k)\widehat{\gamma}_{n}^{-1}(k/(np_{n}))^{\widehat{\gamma}_{n}}$
in the numerator of $\widehat{\pi}_{n}$, whose rate of convergence to
$x^{\star}-q_{1-p_{n}}$ is precisely $\log(k/(np_{n}))/\sqrt{k}$, as obtained
in Proposition 2.

##  3 Simulation study

The finite-sample performance of the proposed extreme expectile estimators is
illustrated here through a simulation study. Our setup first considers three
models for i.i.d. data:

  1. (i)

The $X_{t}$ have a Beta distribution, whose density function is

$f(x|\alpha,\beta)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\text{B}(\alpha,\beta)},\quad
0\leq x\leq 1.$

Here $\text{B}(\alpha,\beta)$ is the Beta function and the shape parameters
are set as $\alpha=3$ and $\beta=5/2$. In this model, the extreme value index
is $\gamma=-2/5$ and the upper endpoint is $x^{*}=1$.

  2. (ii)

The $X_{t}$ have a short-tailed power-law distribution, whose distribution
function is

$F(x|x^{*},K,\alpha)=1-K(x^{*}-x)^{\alpha},\quad x^{*}-K^{-1/\alpha}\leq x\leq
x^{*}.$

Here $x^{\star}$, $K$ and $\alpha$ are the distribution endpoint, a positive
constant and the shape parameter, respectively, which have been set as
$x^{*}=5$, $K=1/3$ and $\alpha=3$, so that $\gamma=-1/3$.

  3. (iii)

The $X_{t}$ have a GEV distribution, whose distribution function is

$F(x|\gamma)=\exp(-(1+\gamma x)^{-1/\gamma}),\quad 1+\gamma x>0.$

We set the extreme value index $\gamma=-1/3$, so that the upper endpoint is
$x^{*}=-1/\gamma=3$.

We then consider the following three time series models, in which $\Phi$
denotes the standard normal distribution function and $Y_{t}$ is the AR(1)
process defined as $Y_{t+1}=\varrho
Y_{t}+\sqrt{1-\varrho^{2}}\,\varepsilon_{t}$, with independent standard normal
innovations $\varepsilon_{t}$, and where $\varrho\in(-1,1)$:

  1. (iv)

$X_{t}=q_{X}(\Phi(Y_{t}))$, where $q_{X}$ is the quantile function
corresponding to the Beta distribution defined in (i), and where the
correlation parameter is $\varrho=0.95$.

  2. (v)

$X_{t}=q_{X}(\Phi(Y_{t}))$, where $q_{X}$ is the quantile function
corresponding to the short-tailed power-law distribution defined in (ii), and
where the correlation parameter is $\varrho=0.5$.

  3. (vi)

$X_{t}=q_{X}(\Phi(Y_{t}))$, where $q_{X}$ is the quantile function
corresponding to the GEV distribution defined in (iii), and where the
correlation parameter is $\varrho=0.8$.

The EVI and endpoints of models (iv)-(vi) are those of models (i)-(iii),
respectively, and the time series models (iv)-(vi) are geometrically
$\beta$-mixing (and in particular geometrically $\alpha-$mixing) since the
linear AR(1) process $(Y_{t})$ is so. We consider sample sizes $n=150,300,500$
and aim to predict expectiles of extreme level
$\tau_{n}^{\prime}=1-p_{n}=1-1/n=0.9933,0.9967,0.9980$. As the true expectile
values cannot be given in closed form, they have been computed by intensive
Monte Carlo simulations and are reported in Table 1.

Model | $\tau^{\prime}_{n}=0.9933$ | $\tau^{\prime}_{n}=0.9967$ | $\tau^{\prime}_{n}=0.9980$
---|---|---|---
(i), (iv) | $0.8571$ | $0.8814$ | $0.8968$
(ii), (v) | $4.5284$ | $4.5939$ | $4.6372$
(iii), (vi) | $1.9523$ | $2.1020$ | $2.2000$
Table 1: Values of the expectile $\xi_{\tau^{\prime}_{n}}$ obtained through
intensive Monte Carlo simulations for $\tau^{\prime}_{n}=1-1/n$, with
$n=150,300,500$.

We simulate $M=10{,}000$ samples of $n$ observations from each model and
compare the purely empirical (LAWS) estimator
$\widehat{\xi}_{\tau_{n}^{\prime}}$ in (2.6), the extrapolating LAWS
estimators $\widehat{\xi}^{\star}_{\tau_{n}^{\prime}}$ in (2.8) obtained by
setting
$\widehat{\sigma}_{n}=\widehat{a}(1/\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}}))$,
its alternative version $\overline{\xi}^{\star}_{\tau_{n}^{\prime}}$ obtained
with
$\widehat{\sigma}_{n}=\widehat{a}((1-\tau_{n})^{-1})\times((1-\tau_{n})/\widehat{\overline{F}}_{n}(\widehat{\xi}_{\tau_{n}}))^{\widehat{\gamma}_{n}}$
in view of the approximation
$a(1/\overline{F}(\xi_{\tau_{n}}))\approx((1-\tau_{n})/\overline{F}(\xi_{\tau_{n}}))^{\gamma}a((1-\tau_{n})^{-1})$
that follows from the regular variation property of the scale function $a$,
and the extrapolating quantile-based (QB) estimator
$\widetilde{\xi}^{\star}_{\tau_{n}^{\prime}}$ in (2.11). In these last three
estimators, $(\widehat{a}(n/k),\widehat{\gamma}_{n})$ are either the pair of
GP-pseudo-ML estimators of $(a(n/k),\gamma)$ (see de Haan and Ferreira,, 2006,
Section 3.4) based on the top $k$ observations in the sample, or their
versions based on the Dekkers et al., (1989) Moment estimator (see de Haan and
Ferreira,, 2006, Section 3.5 and 4.2). We set throughout $\tau_{n}=1-k/n$, let
the effective sample size $k$ range from $1\%$ up to $25\%$ of the total
sample size $n$, and record Monte Carlo approximations of the relative bias,
variance and Mean Squared Error (MSE) of the estimators as a function of $k$.

Results are reported in Figures B.1–B.6 in Section B.1. In each figure the
relative bias, variance and MSE are displayed from left to right, and results
related to sample sizes $n=150,300,500$ are shown from top to bottom. For the
sake of brevity we only report below in Figure 2 the results obtained with the
Beta distribution, for the sample size $n=300$ that we will also consider in
our real data analysis, but we discuss the conclusions from the full set of
models in Section B.1. The Beta model corresponds to a case in which the
extreme value bias is present (unlike in the power-law setting, which is a
transformation of a pure Pareto model) but not too disruptive in small samples
(unlike in the case of the GEV distribution, which should be seen as difficult
from that perspective). On the basis of the bias, the empirical estimator and
extrapolating QB estimator tend to underestimate the true expectile along the
entire range of $k$ values, while the extrapolating LAWS estimator tends to
overestimate the true expectile (at least when the scale and shape parameters
are estimated via GP-pseudo-ML). From the variance point of view, the
extrapolating QB estimator is overall best among the estimators we consider,
with the extrapolating LAWS estimators having large variance for small values
of $k$. Variability of the estimates seems to be highest when the data comes
from time series. This conclusion carries over to the MSE: based on this
criterion, the extrapolating QB estimator overall performs best, as expected
from Remark 12, with the extrapolating LAWS estimator sometimes outperforming
the extrapolating QB estimator for effective sample fractions larger than
$20\%$. In general, both extrapolating QB and LAWS estimators seem to perform
remarkably well relative to the purely empirical expectile estimator, given
the small sample size in this study.

##  4 Application to forecast verification and comparison

In this section, we apply our LAWS and QB methods to estimate tail expectile
risk for Bitcoin (BTC-USD), a peer-to-peer digital decentralized
cryptocurrency. At the end of September 2014, Bitcoin had volatility seven
times greater than gold, eight times greater than the S&P 500, and 18 times
greater than the US dollar. Although the growth of Bitcoin prices has been
often described as an economic bubble, the COVID-19 crisis has sparked
substantial investment in this digital currency as an alternative to
conventional asset classes. Similarly to the 14 major companies and financial
institutions that we have explored in Figure 1, we will provide evidence that
short-tailed returns occur over (relatively) short time periods for this
cryptocurrency as well. To do so and to assess its associated extreme
financial risk accordingly, we construct a time series of weekly negative log-
returns from averaged daily Bitcoin closing prices within the corresponding
week, from September 28th, 2014, to June 12th, 2022. The time series of loss
returns is represented in Figure 3 (A).

We consider risk assessment from a forecasting perspective. With our knowledge
of this week, the goal is to give the best possible point estimate of the
expectile risk measure $\xi_{\tau_{n}^{\prime}}$ for the next week based on
rolling windows of length $n=300$. This results in 103 samples of size $n$
over the observed timeframe. For each sample $(X_{1},\ldots,X_{n})$, the EVI
of the underlying distribution was estimated by means of the ML method for
peaks over a high threshold $X_{n-k,n}$. The plot of the estimates obtained
over the successive 103 rolling windows is given in Figure 3 (B), where an
appropriate $k$ is chosen, for each sample, by regarding the path of the
$\gamma$ estimator as a function of $k$ and selecting the $k$ value which
corresponds to the median estimate over the most stable region of the path
(this can be achieved by using the algorithm developed by El Methni and
Stupfler, (2017)). This selection is highlighted in Figure 3 (B) by a colour
scheme, ranging from dark red (low) to dark violet (high). The final $\gamma$
estimates are found to be all negative in $[-0.147,-0.057]$. It should be
noted that we have comfortably concluded the stationarity of the time series
samples across all $T=103$ rolling windows, from our exploratory analysis.

Expectiles have recently received growing attention in quantitative risk
management not only for their coherence as a tail risk measure, but also for
their elicitability that corresponds to the existence of a natural methodology
for forecast verification. According to Gneiting (2011) and Ziegel (2016),
letting the random variable $X$ model the future observation of interest,
$\xi_{\tau_{n}^{\prime}}$ equals the optimal point forecast for $X$ given by
the Bayes rule
$\xi_{\tau_{n}^{\prime}}=\operatorname*{arg\,min}_{\xi\in\mathbb{R}}\mathbb{E}\left[L_{\tau_{n}^{\prime}}(\xi,X)\right]$,
under the asymmetric quadratic scoring function

$L_{\tau_{n}^{\prime}}:\,\mathbb{R}^{2}\longrightarrow[0,\infty),\quad(\xi,x)\mapsto\eta_{\tau_{n}^{\prime}}(x-\xi),$

where $L_{\tau_{n}^{\prime}}(\xi,x)$ represents the loss or penalty when the
point forecast $\xi$ is issued and the realization $x$ of $X$ materializes.
Following the ideas of Gneiting (2011) and Ziegel (2016), the competing
estimation procedures for $\xi_{\tau_{n}^{\prime}}$ can be compared by using
the scoring function $L_{\tau_{n}^{\prime}}$: Suppose that, in $T$ forecast
cases, we have point forecasts
$\left(\xi^{(m)}_{1},\ldots,\xi^{(m)}_{T}\right)$ and realizing observations
$(x_{1},\ldots,x_{T})$, where the index $m$ numbers the competing forecasters
that are computed at each forecast case $t=1,\ldots,T$. In the assessment, we
compare the purely empirical expectile
$\xi^{(1)}_{t}:=\widehat{\xi}_{\tau_{n}^{\prime}}$ in (2.6) with the direct
extrapolating LAWS estimator
$\xi^{(2)}_{t}:=\widehat{\xi}_{\tau_{n}^{\prime}}^{\star}$ in (2.8) and its
alternative version
$\xi^{(3)}_{t}:=\overline{\xi}_{\tau_{n}^{\prime}}^{\star}$ described in
Section 3, and with the indirect QB extrapolating estimator
$\xi^{(4)}_{t}:=\widetilde{\xi}_{\tau_{n}^{\prime}}^{\star}$ in (2.11), all of
them being based on the GP-pseudo-ML estimators
$(\widehat{a}(n/k),\widehat{\gamma}_{n})$ of $(a(n/k),\gamma)$. When the
Moment estimators $(\widehat{a}(n/k),\widehat{\gamma}_{n})$ are used instead
of the ML estimators, the corresponding three extrapolating forecasters
$\widehat{\xi}_{\tau_{n}^{\prime}}^{\star}$,
$\overline{\xi}_{\tau_{n}^{\prime}}^{\star}$ and
$\widetilde{\xi}_{\tau_{n}^{\prime}}^{\star}$ will be denoted in the sequel by
replacing “$\star$” with “$\clubsuit$” to define
$\xi^{(5)}_{t}:=\widehat{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$,
$\xi^{(6)}_{t}:=\overline{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$ and
$\xi^{(7)}_{t}:=\widetilde{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$. The seven
competing point estimates can then be ranked in terms of their average scores
(the lower the better):

$\overline{L}^{(m)}_{\tau_{n}^{\prime}}=\frac{1}{T}\sum_{t=1}^{T}L_{\tau_{n}^{\prime}}\left(\xi^{(m)}_{t},x_{t}\right),\quad
m=1,\ldots,7.$

The computation of the different extrapolated expectile estimators requires,
like the EVI estimators, the determination of the optimal value of the
effective sample size $k$. By balancing the potential estimation bias and
variance, a usual practice in extreme value theory is to choose $k$ from the
first stable region of the plots [see, e.g., Section 3 in de Haan and Ferreira
(2006)]. This is achieved by using the path stability procedure for $\gamma$
estimation. However, to achieve optimal point forecasts
$\left(\xi^{(m)}_{1},\ldots,\xi^{(m)}_{T}\right)$ for the future observation
$X$, this requires the use of $k$ values that minimize their associated
realized loss
$\overline{L}^{(m)}_{\tau_{n}^{\prime}}\equiv\overline{L}^{(m)}_{\tau_{n}^{\prime}}(k)$,
for m=2,…,7. Doing so, we obtain the final values of
$\overline{L}^{(m)}_{\tau_{n}^{\prime}}$ graphed in Figure 3 (C), as functions
of the extreme level $\tau_{n}^{\prime}\in[0.99,1]$, for the seven competing
estimators. It can be seen that the LAWS-Moment estimator
$\widehat{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$ (dashed magenta) is the best
forecaster uniformly in $\tau_{n}^{\prime}$, followed by the LAWS-ML estimator
$\widehat{\xi}_{\tau_{n}^{\prime}}^{\star}$ (solid magenta) and then the QB-ML
estimator $\widetilde{\xi}_{\tau_{n}^{\prime}}^{\star}$ (solid black). The
remaining three extrapolating estimators do not seem, for this particular
choice of $T=103$ rolling windows of length $n=300$, to outperform the naive
sample expectile $\widehat{\xi}_{\tau_{n}^{\prime}}$ (dashed orange). The
values of the top-ranked forecaster
$\widehat{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$, computed on the 103
successive rolling windows for the extreme levels
$\tau_{n}^{\prime}\in\\{0.99,0.9933,0.9966\\}$, are displayed in Figure 3 (D),
along with the realizing observation at each forecast case. The point
forecasts seem to smoothly increase with $\tau_{n}^{\prime}$ approaching the
worst expected (finite) losses at $\tau_{n}^{\prime}=1$. From the perspective
of pessimistic decision making, the forecasts obtained at the lower level
$\tau_{n}^{\prime}=0.99$ (orange curve) are already cautious since they do lie
almost overall beyond the range of the data: This is mainly due to the short-
tailed nature of Bitcoin data that is closer to light-tailedness.

Extreme expectiles can also serve as a useful tool for estimating the
conventional Value at Risk (VaR) itself. Stated differently, if the
statistician or the practitioner wishes to forecast a coherent expectile
$\xi_{\tau_{n}^{\prime}}$ that has the same probabilistic interpretation as an
extreme quantile $q_{\alpha_{n}}$, for a pre-specified tail probability level
$\alpha_{n}$, a natural way of doing so is to select the asymmetry level
$\tau_{n}^{\prime}$ so that $\xi_{\tau_{n}^{\prime}}\equiv q_{\alpha_{n}}$. As
justified in Section 2.4, such a $\tau_{n}^{\prime}$ can be estimated by
$\widehat{\tau}_{n}^{\prime}=1-\widehat{\pi}_{n}$. When substituting this
estimated value in place of $\tau_{n}^{\prime}$ in our
$\xi_{\tau_{n}^{\prime}}$ extrapolated estimators, the latter estimate the VaR
$q_{\alpha_{n}}$ itself and can then be compared with the popular GP fit
$\widehat{q}_{\alpha_{n}}^{\star}$ defined in (2.10). Here also, forecast
verification and comparison is possible thanks to the elicitability property
of quantiles (see e.g. Gneiting (2011)). Given that it is the quantile level
$\alpha_{n}$ which is fixed in advance, the accuracy of the associated VaR
forecasts is to be assessed by means of the realized loss

$\overline{L}^{(m)}_{\alpha_{n}}=\frac{1}{T}\sum_{t=1}^{T}L_{\alpha_{n}}\left(q^{(m)}_{t},x_{t}\right),\quad
m=1,\ldots,8,$

under the asymmetric piecewise linear scoring function

$L_{\alpha_{n}}:\,\mathbb{R}^{2}\longrightarrow[0,\infty),\quad(q,x)\mapsto\varrho_{\alpha_{n}}(x-q),$

for the competing ML-based forecasters
$q^{(1)}_{t}:=\widehat{q}_{\alpha_{n}}^{\star}$,
$q^{(2)}_{t}:=\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$,
$q^{(3)}_{t}:=\overline{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$,
$q^{(4)}_{t}:=\widetilde{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$, and
their Moment-based versions
$q^{(5)}_{t}:=\widehat{q}_{\alpha_{n}}^{\clubsuit}$,
$q^{(6)}_{t}:=\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$,
$q^{(7)}_{t}:=\overline{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$, and
$q^{(8)}_{t}:=\widetilde{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$. The
resulting realized losses $\overline{L}^{(m)}_{\alpha_{n}}$ are graphed in
Figure 3 (E), as functions of the quantile level $\alpha_{n}\in[0.99,1]$, for
the eight competing estimators of
$q_{\alpha_{n}}\equiv\xi_{\tau_{n}^{\prime}}$. It is remarkable that the best
forecaster is still the LAWS-Moment estimator
$\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$ (dashed magenta),
followed by the LAWS-ML estimator
$\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$ (solid magenta). Most
importantly, these expectile-based forecastors clearly outperform the usual
GP-ML fit $\widehat{q}_{\alpha_{n}}^{\star}$ (solid orange) and GP-Moment fit
$\widehat{q}_{\alpha_{n}}^{\clubsuit}$ (dashed orange), which is good news to
practitioners whose concern is to assess the accuracy of forecasts. Figure 3
(F) contrasts the evolution of the optimal point forecasts
$\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$, for the risk
measure $q_{\alpha_{n}}$ at the extreme levels
$\alpha_{n}\in\\{0.99,0.9933,0.9966\\}$, with the realizing observations. By
comparing these $\alpha_{n}$th quantile estimates with their expectile analogs
from Figure 3 (D) at the same asymmetry levels
($\alpha_{n}=\tau_{n}^{\prime}$), it may be seen that expectiles are
ultimately less conservative than quantiles, which empirically corroborates
the theoretical result for short-tailed data in Proposition 2.2 by Bellini and
Di Bernardino, (2017). This more liberal expectile assessment of tail risk is
indeed a consequence of the diversification principle satisfied by expectiles.
Interestingly, the conservative LAWS-Moment (expectile-based) forecasts
$\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$, for
$q_{\alpha_{n}}$ in Figure 3 (F), seem also to be more sensitive to the
variability of weekly losses compared with their analog forecasts
$\widehat{\xi}_{\alpha_{n}}^{\clubsuit}$ for $\xi_{\alpha_{n}}$ in Figure 3
(D).

Figure 2: Empirical relative bias, variance and MSE (left, middle and right),
multiplied by $100$, for the $\xi_{\tau_{n}^{\prime}}$ estimators obtained
with observations from a Beta distribution, $\tau_{n}^{\prime}=1-1/n$ and
$n=300$. Empirical estimator $\widehat{\xi}_{\tau^{\prime}_{n}}$ (orange
line), extrapolating LAWS estimators
$\widehat{\xi}^{\star}_{\tau^{\prime}_{n}}$ (magenta lines) and
$\overline{\xi}^{\star}_{\tau^{\prime}_{n}}$ (blue lines), and extrapolating
QB estimators $\widetilde{\xi}^{\star}_{\tau^{\prime}_{n}}$ (black lines). The
extrapolating estimators based on the scale and shape parameter estimates from
the GP-pseudo-ML are referred to using solid lines, and those based on the
Moment estimators are referred to using dashed lines. Top: i.i.d. data,
bottom: nonlinear AR(1) data.

Figure 3:  (A) Bitcoin weekly loss returns from September 28th, 2014, to June
12th, 2022. (B) GP-pseudo-ML estimates of $\gamma$ over the 103 rolling
windows. (C) Realized loss function
$\tau_{n}^{\prime}\mapsto\overline{L}^{(m)}_{\tau_{n}^{\prime}}$ evaluated at
$\tau_{n}^{\prime}\in[0.99,1]$ for the seven competing forecasters
$\xi^{(1)}_{t}=\widehat{\xi}_{\tau_{n}^{\prime}}$ (orange),
$\xi^{(2)}_{t}=\widehat{\xi}_{\tau_{n}^{\prime}}^{\star}$ (solid magenta),
$\xi^{(3)}_{t}=\overline{\xi}_{\tau_{n}^{\prime}}^{\star}$ (solid blue),
$\xi^{(4)}_{t}=\widetilde{\xi}_{\tau_{n}^{\prime}}^{\star}$ (solid black),
$\xi^{(5)}_{t}=\widehat{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$ (dashed
magenta), $\xi^{(6)}_{t}=\overline{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$
(dashed blue), and
$\xi^{(7)}_{t}=\widetilde{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$ (dashed
black). (D) The top-ranked optimal point forecaster
$\widehat{\xi}_{\tau_{n}^{\prime}}^{\clubsuit}$ of the risk measure
$\xi_{\tau_{n}^{\prime}}$, for $\tau_{n}^{\prime}=0.99$ (orange),
$\tau_{n}^{\prime}=0.9933$ (magenta) and $\tau_{n}^{\prime}=0.9966$ (blue),
along with the realizing observations (gray). (E) Realized loss function
$\alpha_{n}\mapsto\overline{L}^{(m)}_{\alpha_{n}}$ evaluated at
$\alpha_{n}\in[0.99,1]$ for the eight competing forecasters
$q^{(1)}_{t}:=\widehat{q}_{\alpha_{n}}^{\star}$ (solid orange),
$q^{(2)}_{t}:=\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$ (solid
magenta), $q^{(3)}_{t}:=\overline{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$
(solid blue),
$q^{(4)}_{t}:=\widetilde{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\star}$ (solid
black), $q^{(5)}_{t}:=\widehat{q}_{\alpha_{n}}^{\clubsuit}$ (dashed orange),
$q^{(6)}_{t}:=\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$ (dashed
magenta),
$q^{(7)}_{t}:=\overline{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$
(dashed blue), and
$q^{(8)}_{t}:=\widetilde{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$
(dashed black). (F) The top-ranked optimal point forecaster
$\widehat{\xi}_{\widehat{\tau}_{n}^{\prime}}^{\clubsuit}$ of the risk measure
$q_{\alpha_{n}}$, for $\alpha_{n}=0.99$ (orange), $\alpha_{n}=0.9933$
(magenta) and $\alpha_{n}=0.9966$ (blue), along with the realizing
observations (gray).

## Acknowledgments

This research was supported by the French National Research Agency under the
grants ANR-19-CE40-0013 and ANR-17-EURE-0010. S.A. Padoan is supported by the
Bocconi Institute for Data Science and Analytics (BIDSA), Italy. A. Daouia and
G. Stupfler acknowledge financial support from the TSE-HEC ACPR Chair and from
an AXA Research Fund Award on “Mitigating risk in the wake of the COVID-19
pandemic”.

## References

  * Beirlant et al., (2004) Beirlant, J., Goegebeur, Y., Segers, J., and Teugels, J. (2004).  Statistics of Extremes: Theory and Applications.  Wiley.
  * Bellini and Di Bernardino, (2017) Bellini, F. and Di Bernardino, E. (2017).  Risk management with expectiles.  The European Journal of Finance, 23(6):487–506.
  * Bellini et al., (2014) Bellini, F., Klar, B., Müller, A., and Gianin, E. R. (2014).  Generalized quantiles as risk measures.  Insurance: Mathematics and Economics, 54:41–48.
  * Bradley, (2005) Bradley, R. C. (2005).  Basic properties of strong mixing conditions. A survey and some open questions.  Probability Surveys, 2:107–144.
  * Daouia et al., (2018) Daouia, A., Girard, S., and Stupfler, G. (2018).  Estimation of tail risk based on extreme expectiles.  Journal of the Royal Statistical Society: Series B, 80(2):263–292.
  * Daouia et al., (2020) Daouia, A., Girard, S., and Stupfler, G. (2020).  Tail expectile process and risk assessment.  Bernoulli, 26(1):531–556.
  * Davison et al., (2022) Davison, A. C., Padoan, S. A., and Stupfler, G. (2022).  Tail risk inference via expectiles in heavy-tailed time series.  Journal of Business and Economic Statistics, to appear, available at https://doi.org/10.1080/07350015.2022.2078332.
  * de Haan and Ferreira, (2006) de Haan, L. and Ferreira, A. (2006).  Extreme Value Theory: An Introduction.  Springer-Verlag, New York.
  * Dekkers et al., (1989) Dekkers, A., Einmahl, J., and de Haan, L. (1989).  A moment estimator for the index of an extreme-value distribution.  Annals of Statistics, 17(4):1833–1855.
  * Drees, (2003) Drees, H. (2003).  Extreme quantile estimation for dependent data, with applications to finance.  Bernoulli, 9(4):617–657.
  * El Methni and Stupfler, (2017) El Methni, J. and Stupfler, G. (2017).  Extreme versions of wang risk measures and their estimation for heavy-tailed distributions.  Statistica Sinica, 27(2):907–930.
  * Girard et al., (2021) Girard, S., Stupfler, G., and Usseglio-Carleve, A. (2021).  Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models.  Annals of Statistics, 49(6):3358–3382.
  * Girard et al., (2022) Girard, S., Stupfler, G., and Usseglio-Carleve, A. (2022).  On automatic bias reduction for extreme expectile estimation.  Statistics & Computing, 32(4):64.
  * Gneiting, (2011) Gneiting, T. (2011).  Making and evaluating point forecasts.  Journal of the American Statistical Association, 106(494):746–762.
  * Hjort and Pollard, (1993) Hjort, N. L. and Pollard, D. (1993).  Asymptotics for minimisers of convex processes.  Statistical Research Report, University of Oslo.
  * Jones, (1994) Jones, M. C. (1994).  Expectiles and M-quantiles are quantiles.  Statistics $\&$ Probability Letters, 20(2):149–153.
  * Koenker and Bassett, (1978) Koenker, R. and Bassett, G. (1978).  Regression quantiles.  Econometrica, 46(1):33–50.
  * Kokic et al., (1997) Kokic, P., Chambers, R., Breckling, J., and Beare, S. (1997).  A measure of production performance.  Journal of Business & Economic Statistics, 15(4):445–451.
  * Kuan et al., (2009) Kuan, C.-M., Yeh, J.-H., and Hsu, Y.-C. (2009).  Assessing value at risk with CARE, the Conditional Autoregressive Expectile models.  Journal of Econometrics, 150(2):261–270.
  * Mao et al., (2015) Mao, T., Ng, K., and Hu, T. (2015).  Asymptotic expansions of generalized quantiles and expectiles for extreme risks.  Probability in the Engineering and Informational Sciences, 29(3):309–327.
  * Newey and Powell, (1987) Newey, W. K. and Powell, J. L. (1987).  Asymmetric least squares estimation and testing.  Econometrica, 55(4):819–847.
  * Padoan and Stupfler, (2022) Padoan, S. A. and Stupfler, G. (2022).  Joint inference on extreme expectiles for multivariate heavy-tailed distributions.  Bernoulli, 28(2):1021–1048.
  * Rootzén et al., (1998) Rootzén, H., Leadbetter, M. R., and de Haan, L. (1998).  On the distribution of tail array sums for strongly mixing stationary sequences.  Annals of Applied Probability, 8(3):868–885.
  * Schmidt and Stadtmüller, (2006) Schmidt, R. and Stadtmüller, U. (2006).  Non-parametric estimation of tail dependence.  Scandinavian Journal of Statistics, 33(2):307–335.
  * Taylor, (2008) Taylor, J. (2008).  Estimating Value at Risk and Expected Shortfall using expectiles.  Journal of Financial Econometrics, 6(2):231–252.
  * Ziegel, (2016) Ziegel, J. F. (2016).  Coherence and elicitability.  Mathematical Finance, 26(4):901–918.