# Statistical depth and support medians for fuzzy data

LUIS GONZÁLEZ-DE LA<EMAIL_ADDRESS>[ ALICIA NIETO-
<EMAIL_ADDRESS>Corresponding author, ORCID:
0000-0002-0268-3322[ PEDRO<EMAIL_ADDRESS>[ Departamento
de Matemáticas, Estadística y Computación, Universidad de Cantabria, Spain
Universidad de Oviedo, Spain

###### Abstract

Statistical depth functions order the elements of a space with respect to
their centrality in a probability distribution or dataset. Since many depth
functions are maximized in the real line by the median, they provide a natural
approach to defining median-like location estimators for more general types of
data (in our case, fuzzy data). We analyze the relationships between depth-
based medians, medians based on the support function, and some notions of a
median for fuzzy data in the literature. We take advantage of specific depth
functions for fuzzy data defined in our former papers: adaptations of Tukey
depth, simplicial depth, $L^{1}$-depth and projection depth.

Median,

Fuzzy random variable,

Nonparametric statistics,

Statistical depth,

Fuzzy data,

###### keywords:

, and

##  1 Introduction

Statistical depth functions order the elements of a space with respect to a
distribution on the space, and, specifically, the elements of a dataset. This
order is trivial in dimension one, providing the concept of quantiles and, in
particular, of median. However, it is not trivial in higher dimensions. A way
to see it is that there exist multiple manners of extending the idea of
median, and other quantiles, to spaces of dimension larger than one while
satisfying certain desirable properties. These desirable properties are
gathered in axiomatic notions of statistical depth, see [43] for multivariate
spaces, [31] for functional, metric, spaces and [13] for spaces of fuzzy sets.
Each notion takes into account the particular characteristics of the
underlying space. Examples of depth functions are: Tukey depth or halfspace
depth [38], random Tukey depth [4] and simplicial depth [22] in multivariate
spaces, spatial depth [3], metric depth [30] and lens metric depth [16] in
functional spaces and fuzzy Tukey depth [13], fuzzy simplicial-type depths
[14], fuzzy projection depth [15] and fuzzy $L^{1}$-type depth [15] in fuzzy
spaces.

A naive but straightforward way of computing quantiles, in, for instance,
multivariate spaces, is the component-wise quantiles, and, in particular, the
component-wise median. However, this has some undesired properties: (I) The
coordinate-wise median of a dataset does not necessarily belong to the convex
hull of the dataset [32]. (II) When computed with respect to the vertices of
an equilateral triangle and its center of mass, the component-wise median
depends on the coordinate axis and it is not necessarily the center of mass
[9]. This undesired behavior of naive strategies has resulted in that lately
the median induced by the different depth functions is used in the literature
to generalized the, one-dimensional, median. Examples are [42], [39] or [23]
in multivariate spaces and [24], [27] or [10] in functional spaces. This paper
is dedicated to propose and study, among others, the medians induced by the
existing fuzzy depth functions.

In the fuzzy setting, the literature contains already some notions of median
and quantile functions of fuzzy random variables, which are not induced in
their conceptualization by stiatitical fuzzy depth functions. Each of them is
defined for the specific case of $\mathcal{F}_{c}(\mathbb{R})$. In [17],
Grzegorzewski proposes the definition of a median of a fuzzy random variable
based on the Zadeh’s extension principle. More recently, Sinova et al. [35]
propose the notion of median of a fuzzy random variable (the 1-median) from an
$L^{1}$-minimization principle. Also Shvedov in [34] proposes the definition
of a quantile function of a fuzzy random variable, which assigns to every
$p\in(0,1)$ a fuzzy set. In that setting, if we take $p=1/2$, we obtain the
median of the fuzzy random variable. In the present paper we focus our study
to the concepts of medians proposed by Sinova et al. [35] as a minimization
problem and by Grzegorzewski [17] using the Zadeh’s extension principle and
connect them with our proposals. This is due to these proposal are more
consistent with the fuzzy theory than the median in [34], which is a
generalization of the univariate case.

The paper is organized as follows. Section 2 contains the notation and
background, on fuzzy sets and depth, necessary for a comprehensive
understanding of the paper. In Section 3 we present two definitions of fuzzy
medians, one based on support functions and the other on maximizers of depth
functions. A study about the relation between them and with the existing
notions is pursued in Sections 3 and 4. All proofs and further results are
contained in Section 5. A discussion is in Section 6.

##  2 Preliminaries

The following notation is used throughout. We denote by
$\mathcal{K}_{c}(\mathbb{R}^{p})$ the set of all non-empty, compact and convex
subsets of $\mathbb{R}^{p}$. A fuzzy set on $\mathbb{R}^{p}$ is any function
$A\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{p}\rightarrow[0,1]$. The
$\alpha$-levels, or $\alpha$-cuts, of a fuzzy set $A$ are defined as the sets
$A_{\alpha}=\\{x\in\mathbb{R}^{p}\mathrel{\mathop{\mathchar
58\relax}}A(x)\geq\alpha\\}$ with $\alpha\in(0,1]$ and
$A_{0}=\text{clo}(\\{x\in\mathbb{R}^{p}\mathrel{\mathop{\mathchar
58\relax}}A(x)>0\\})$, where $\text{clo}(\cdot)$ denotes the closure of a set.
The set of fuzzy sets whose $\alpha$-levels are elements of
$\mathcal{K}_{c}(\mathbb{R}^{p})$ is denoted by
$\mathcal{F}_{c}(\mathbb{R}^{p})$. Abusing of nomenclature, we refer to the
elements of $\mathcal{F}_{c}(\mathbb{R}^{p})$ as fuzzy sets.

Ordinary sets, $K\in\mathcal{K}_{c}(\mathbb{R}^{p})$, can be identified with
its indicator function, $\text{I}_{K}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{p}\rightarrow\mathbb{R}$, where $\text{I}_{K}(x)=1$ if
$x\in K$ and $\text{I}_{K}(x)=0$ otherwise. The sets of the form
$\text{I}_{K}$ are called crisps sets. Thus, abusing again of the notation, we
consider that
$\mathcal{K}_{c}(\mathbb{R}^{p})\subseteq\mathcal{F}_{c}(\mathbb{R}^{p})$.

Let us denote by $\mathbb{S}^{p-1}\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathbb{R}^{p}\mathrel{\mathop{\mathchar 58\relax}}\|x\|\leq
1\\}$ the unit sphere on $\mathbb{R}^{p}$, with $\|.\|$ the euclidean norm on
$\mathbb{R}^{p}$ and by $\langle\cdot,\cdot\rangle$ the usual inner product on
$\mathbb{R}^{p}$. The mapping $s_{A}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{S}^{p-1}\times[0,1]\rightarrow\mathbb{R}$ defined as
$s_{A}(u,\alpha)=\sup\\{\langle u,v\rangle\mathrel{\mathop{\mathchar
58\relax}}v\in A_{\alpha}\\}$ for every $u\in\mathbb{S}^{p-1}$ and
$\alpha\in[0,1]$ is known as the support function of the fuzzy set
$A\in\mathcal{F}_{c}(\mathbb{R}^{p})$.

It is common to consider triangular fuzzy sets in the case of
$\mathcal{F}_{c}(\mathbb{R})$ (see [18, Section 4.1]), also known as
triangular fuzzy numbers. Let $a,b,c\in\mathbb{R}$ with $a\leq b\leq c,$ the
triangular fuzzy number $T(a,b,c)$ is given by

$T(a,b,c)(x)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{lcr}\cfrac{x-a}{b-a}&\text{ if
}x\in[a,b]\text{ and }a<b,\\\ \cfrac{x-c}{b-c}&\text{ if }x\in[b,c]\text{ and
}b<c\\\ 1&\text{ if }x\in[a,b]\text{ and }a=b\text{ or }x\in[b,c]\text{ and
}b=x,\\\ 0&\text{otherwise.}\end{array}\right.$

###  2.1 Zadeh’s extension principle

It is possible to define new fuzzy sets from non-fuzzy/crisp functions acting
over fuzzy sets, using the Zadeh’s extension principle [41]. Given
$A\in\mathcal{F}_{c}(\mathbb{R}^{p})$ and a function
$f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{p}\rightarrow\mathbb{R}^{p},$ the image $f(A)$ is
defined to be $f(A)(t)=\sup\\{A(y)\mathrel{\mathop{\mathchar
58\relax}}y\in\mathbb{R}^{p},f(y)=t\\}$ for all $t\in\mathbb{R}^{p}$. If we
consider an arbitrary $f$, it could happen that the set $f(A)$ does not belong
to $\mathcal{F}_{c}(\mathbb{R}^{p})$. Thus we only consider continuous
functions.

###  2.2 Metrics in the fuzzy setting

It is possible to endow $\mathcal{F}_{c}(\mathbb{R}^{p})$ of a metric space
structure, using for instance the well-known familiy of $\rho_{r}$ metrics,
which are $L^{r}$-type metrics (see [7]). The $\rho_{r}$ metric for any
$A,B\in\mathcal{F}_{c}(\mathbb{R}^{p})$ is defined as

$\rho_{r}(A,B)\mathrel{\mathop{\mathchar
58\relax}}=\left(\int_{\mathbb{S}^{p-1}}\int_{[0,1]}\|s_{A}(u,\alpha)-s_{B}(u,\alpha)\|^{r}d\mathcal{V}^{p}(u)\,d\nu(\alpha)\right)^{1/r},$
(1)

where $\nu$ represents the Lebesgue measure and $\mathcal{V}_{p}$ denotes the
normalized Haar measure in $\mathbb{S}^{p-1}$.

###  2.3 Fuzzy random variables

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space, a random compact
set is defined as a function $\Gamma\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{K}_{c}(\mathbb{R}^{p})$ such that the set
$\\{\omega\in\Omega\mathrel{\mathop{\mathchar 58\relax}}\Gamma(\omega)\cap
K\neq\emptyset\\}$ is an element of $\mathcal{A}$, whenever
$K\in\mathcal{K}_{c}(\mathbb{R}^{p})$ (see Molchanov [26]). Based on Puri and
Ralescu [33] a fuzzy random variable is defined as a function
$\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{F}_{c}(\mathbb{R}^{p})$ such that
$\mathcal{X}_{\alpha}(\omega)$ is a random compact set for all
$\alpha\in[0,1]$, where $\mathcal{X}_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{P}(\mathbb{R}^{p})$ is defined as the
$\alpha$-cut of every image of $\mathcal{X}$, i.e.
$\mathcal{X}_{\alpha}(\omega)\mathrel{\mathop{\mathchar
58\relax}}=\\{x\in\mathbb{R}^{p}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}(\omega)(x)\geq\alpha\\}$ for any $\omega\in\Omega$. In
the case of $p=1$, we refer to the fuzzy random variables as fuzzy random
numbers.

The class of fuzzy random variables $\mathcal{X}$ on the measurable space
$(\Omega,\mathcal{A})$ such that $\text{E}[\|\mathcal{X}_{0}\|^{r}]<\infty$
whenever $r\in[1,\infty)$ is denoted by
$L^{r}[\mathcal{F}_{c}(\mathbb{R})^{p}]$. Thus,
$L^{0}[\mathcal{F}_{c}(\mathbb{R})^{p}]$ is the more general class of fuzzy
random variables.

Given $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})^{p}],$ its support
function $s_{\mathcal{X}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{S}^{p-1}\times[0,1]\times\Omega\rightarrow\mathbb{R}$ is
defined by $s_{\mathcal{X}}(u,\alpha,\omega)\mathrel{\mathop{\mathchar
58\relax}}=s_{\mathcal{X}(\omega)}(u,\alpha).$

The function $s_{\mathcal{X}}(u,\alpha)\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathbb{R}$ is a real random variable whenever
$u\in\mathbb{S}^{p-1}$ and $\alpha\in[0,1]$ [1]. We denote by $F_{u,\alpha}$
the distribution function of $s_{\mathcal{X}}(u,\alpha)$.

###  2.4 Depth functions for fuzzy sets

Let $\mathcal{H}\subseteq L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ and
$\mathcal{J}\subseteq\mathcal{F}_{c}(\mathbb{R}^{p}).$ The following depths
are functions of the type $D\mathrel{\mathop{\mathchar
58\relax}}\mathcal{J}\times\mathcal{H}\rightarrow[0,\infty)$ evaluated at a
fuzzy set $A\in\mathcal{J}$ with respect to a fuzzy random variable
$\mathcal{X}\in\mathcal{H}.$ For any $r\in[1,\infty)$, the r-natural depth
[15] is

$D_{r}(A;\mathcal{X})=(1+\text{E}[\rho_{r}(A;\mathcal{X})])^{-1}$ (2)

with $D_{r}(A;\mathcal{X})=0$ in case of expectation being infinite. The Tukey
depth [13] is defined as

$D_{FT}(A;\mathcal{X})=\inf_{u\in\mathbb{S}^{p-1},\alpha\in[0,1]}\min(\mathbb{P}[\omega\in\Omega\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}(\omega)\in
S_{u,\alpha}^{-}],\mathbb{P}[\omega\in\Omega\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}(\omega)\in S_{u,\alpha}^{+}]),$ (3)

where
$S_{u,\alpha}^{-}=\\{U\in\mathcal{F}_{c}(\mathbb{R}^{p})\mathrel{\mathop{\mathchar
58\relax}}s_{U}(u,\alpha)-s_{A}(u,\alpha)\leq 0\\}$ and
$S_{u,\alpha}^{+}=\\{U\in\mathcal{F}_{c}(\mathbb{R}^{p})\mathrel{\mathop{\mathchar
58\relax}}s_{U}(u,\alpha)-s_{A}(u,\alpha)\geq 0\\}.$ The projection depth
function [15] is defined as $D_{FP}(A;\mathcal{X})=(1+O(A;\mathcal{X}))^{-1}$,
where

$O(A;\mathcal{X})=\sup_{(u,\alpha)\in\mathbb{S}^{p-1}\times[0,1]}\cfrac{|s_{A}(u,\alpha)-\operatorname{med}(s_{\mathcal{X}}(u,\alpha))|}{\text{MAD}(s_{\mathcal{X}}(u,\alpha))}.$
(4)

$\operatorname{med}(\cdot)$ denotes the median of a real random variable, with
the usual convention of taking the mid-point of the interval of medians when
the median is not unique, and $\text{MAD}(\cdot)$ the corresponding median
absolute deviation.

Throughout the paper, for both crisp and fuzzy deta we will follow this
criterion: medians which are defined so as to be unique will be represented by
$\operatorname{med}$ while sets of medians which may potentially include more
than one point will be represented by $\operatorname{Med}$. Thus
$\operatorname{Med}(X)$ represents the interval
$[\underline{\operatorname{med}}(X),\overline{\operatorname{med}}(X)]$ formed
by all medians of a random variable $X$. If the median of $X$ is unique, then
$\operatorname{Med}(X)=\\{\operatorname{med}(X)\\}$. This convention will be
very useful since several medians for fuzzy data will be considered.

The modified simplicial depth [14] is defined as

$D_{mS}(A;\mathcal{X})=\text{E}(\mathcal{V}_{p}\otimes\nu\\{(u,\alpha)\in\mathbb{S}^{p-1}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}s_{A}(u,\alpha)\in[m(u,\alpha),M(u,\alpha)]\\}),$ (5)

where $\mathcal{X}_{1},\ldots,\mathcal{X}_{p+1}$ are independent and
identically distributed random variables distributed as $\mathcal{X}$,
$m(u,\alpha)=\min\\{\mathcal{X}_{1},\ldots,\mathcal{X}_{p+1}\\}$ and
$M(u,\alpha)=\max\\{\mathcal{X}_{1},\ldots,\mathcal{X}_{p+1}\\}$. The fuzzy
simplicial depth [14] is

$D_{FS}(A;\mathcal{X})=\inf_{u\in\mathbb{S}^{p-1}}\text{E}(\nu\\{(u,\alpha)\in\mathbb{S}^{p-1}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}s_{A}(u,\alpha)\in[m(u,\alpha),M(u,\alpha)]\\}).$ (6)

##  3 Definition of support medians and depth medians

It has long been recognized that a sample of ordered linguistic values modeled
by fuzzy sets admits a median (see, e.g., [28]). However, that is just the
median of the linguistic values themselves when regarded as an ordinal
variable. It is interesting to have ways to compute a median when the fuzzy
values in the sample are not obviously ordered or do not reflect an order
known in advance; specially, given the ever-increasing list of methods for
ranking fuzzy numbers (see, e.g., [2, 6, 29]).

In this section, we introduce two proposals to find plausible candidates for
the median of a fuzzy random variable: support medians and depth medians.
Later in the section, we present depth medians as the maximizers of a
statistical depth function. Support medians include the medians defined by
Grzegorzewski [17] and Sinova et al. [35]. Thus, we provide these definitions,
before introducing the support median notion.

Kruse and Meyer [21] introduced the idea of a fuzzy parameter of a fuzzy
random variable. In that setting, Grzegorzewski [17] defined the median as the
perception of the median of a fuzzy random variable. Formally, a Grzegorzewski
median of $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})]$ is a fuzzy
number, $\operatorname{med}_{Gr}(\mathcal{X})$, such that for every
$t\in\mathbb{R}$, its membership functions fulfills

$\operatorname{med}_{Gr}(\mathcal{X})(t)=\sup\\{\inf_{\omega\in\Omega}\mathcal{X}(\omega)(X(\omega))\mathrel{\mathop{\mathchar
58\relax}}X\in\mathcal{B}_{\mathcal{X}},t\in\operatorname{med}(X)\\}.$ (7)

There, $\mathcal{B}_{\mathcal{X}}$ is the set of Borel-measurable mappings
$X\mathrel{\mathop{\mathchar
58\relax}}(\Omega,\mathcal{A})\rightarrow(\mathbb{R},\beta(\mathbb{R}))$,
where $(\Omega,\mathcal{A},\mathbb{P})$ is the probabilistic space associated
with the fuzzy random variable, $\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{F}_{c}(\mathbb{R}),$ and
$\beta(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}.$

Grzegorzewski [17] proved a useful characterization: The fuzzy median
$\operatorname{med}_{Gr}(\mathcal{X})$ is uniquely given by

$(\operatorname{med}_{Gr}(\mathcal{X}))_{\alpha}=[\underline{\operatorname{med}}(\inf\mathcal{X}_{\alpha}),\overline{\operatorname{med}}(\sup\mathcal{X}_{\alpha})]$
(8)

for every $\alpha\in[0,1]$.

As Grzegorzewski’s median is defined using Zadeh’s extension principle, it is
unique. However, the one dimensional median is not necesarily unique.
Additionally, if we define the median as the fuzzy number which maximizes a
certain function measuring central tendency, we may obtain non-unique medians.
Following this idea, a definition for the median of a fuzzy random number is
proposed by Sinova et al. [35].

The 1-median of $\mathcal{X}\in L^{1}[\mathcal{F}_{c}(\mathbb{R})]$ is the set
of minimizers

$\displaystyle\operatorname{Med}_{1}(\mathcal{X})=\arg\min_{U\in\mathcal{F}_{c}(\mathbb{R})}\text{E}[\rho_{1}(U,\mathcal{X})].$
(9)

Each element of $\operatorname{Med}_{1}(\mathcal{X})$ is also called a
$1$-median. Sinova et al. [35] show that in ${\mathcal{F}_{c}}({\mathbb{R}})$
there always exist $1$-medians, by constructing a fuzzy set
$\operatorname{med}_{Si}(\mathcal{X})$ with the property

$s_{\operatorname{med}_{Si}(\mathcal{X})}(u,\alpha)=\operatorname{med}(s_{\mathcal{X}}(u,\alpha))\quad\forall
u\in\mathbb{S}^{0},\alpha\in[0,1].$

Since an element of ${\mathcal{F}_{c}}({\mathbb{R}})$ is identified by its
support function, $\operatorname{med}_{Si}(\mathcal{X})$ is unique in having
that property. Moreover, [35, Theorem 4.1.] states that

$\displaystyle\operatorname{med}_{Si}(\mathcal{X})\in\operatorname{Med}_{1}(\mathcal{X}).$
(10)

It is hard to say that $\operatorname{med}_{Gr}$ is better or worse than
$\operatorname{med}_{Si}$. In some cases, the Grzegorzewski median does not
behave like the ordinary (midpoint) median in ${\mathbb{R}}$. For example, the
median of $\text{I}_{\\{0\\}}$ and $\text{I}_{\\{2\\}}$, which represent the
crisp points $0$ and $2$, is not $\text{I}_{\\{1\\}}$ but $\text{I}_{[0,2]}$.
That is a consequence of the definition via Zadeh’s extension principle,
whence $\operatorname{med}_{Gr}(\mathcal{X})(x)$ represents the degree to
which $x$ is a median of $\mathcal{X}$. In the example, every point between 0
and 2 can fully be considered a median and from this point of view the result
is sensible. On the other hand, in this example $\operatorname{med}_{Si}$ is
$\text{I}_{\\{1\\}}$, which replicates the behaviour of the (midpoint) median
in ${\mathbb{R}}$ as a unique representative. It can also be said that the
information about other medians captured by $\operatorname{med}_{Gr}$ is lost.

Let us introduce now the notion of support median.

######  Definition 3.1.

A fuzzy set $A\in{\mathcal{F}_{c}}({\mathbb{R}}^{p})$ is a support median of a
fuzzy random variable $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$
if $s_{A}(u,\alpha)$ is a median of $s_{\mathcal{X}}(u,\alpha)$ for all
$u\in\mathbb{S}^{p-1}$ and $\alpha\in[0,1].$ That is, if
$s_{A}(u,\alpha)\in\operatorname{med}(s_{\mathcal{X}}(u,\alpha))$ for all
$u\in\mathbb{S}^{p-1}$ and $\alpha\in[0,1]$. The set of the support medians of
$\mathcal{X}$ is denoted $\operatorname{Med}_{s}(\mathcal{X})$.

With the results of Sinova et al. [35], it is easily obtained for
$\mathcal{X}\in L^{1}[\mathcal{F}_{c}(\mathbb{R})]$ that

$\displaystyle\operatorname{med}_{Si}(\mathcal{X})\in\operatorname{Med}_{s}(\mathcal{X}).$
(11)

For $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})]$ it is also easy to
check

$\displaystyle\operatorname{med}_{Gr}(\mathcal{X})\in\operatorname{Med}_{s}(\mathcal{X}).$
(12)

Moreover, one has the following result where we have that each $\alpha$-level
of a support median is contained in the corresponding $\alpha$-level of the
Grzegorzewski fuzzy median.

######  Proposition 3.2.

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})].$ Then,
$A_{\alpha}\subseteq(\text{med}_{Gr})_{\alpha}$ for every
$A\in\operatorname{Med}_{s}(\mathcal{X})$ and $\alpha\in[0,1].$

Thus $\operatorname{med}_{Gr}(\mathcal{X})$ has the maximality property that
each support median of $\mathcal{X}$ must be contained in it.

The results in (11), (12) and Proposition 3.2 hold only for the one-
dimensional ($p=1$) case. For $p>1$, there exist fuzzy random variables
without any support median, as we show in the following example.

######  Example 3.3.

Given the probability space
$(\\{\omega_{1},\omega_{2},\omega_{3}\\},\mathcal{P}(\\{\omega_{1},\omega_{2},\omega_{3}\\}),\mathbb{P})$,
with
$\mathbb{P}(\\{\omega_{1}\\})=\mathbb{P}(\\{\omega_{2}\\})=\mathbb{P}(\\{\omega_{3}\\})$,
let $\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\\{\omega_{1},\omega_{2},\omega_{3}\\}\rightarrow\mathcal{F}_{c}(\mathbb{R}^{2})$
be a fuzzy random variable defined by
$\mathcal{X}(\omega_{1})=\text{I}_{\\{(0,-1)\\}}$,
$\mathcal{X}(\omega_{2})=\text{I}_{\\{(2,0)\\}}$ and
$\mathcal{X}(\omega_{3})=\text{I}_{\\{(1,1)\\}}$.

This example presents that the fuzzy random variable $\mathcal{X}$ has no
support medians, that is, $\operatorname{Med}_{s}(\mathcal{X})=\emptyset$.
Assuming for a contradiction that
$\operatorname{Med}_{s}(\mathcal{X})\neq\emptyset$, let $A$ be a support
median of $\mathcal{X},$ i.e.

$s_{A}(u,\alpha)\in\operatorname{med}(s_{\mathcal{X}}(u,\alpha))\mbox{ for all
}u\in\mathbb{S}^{p-1}\mbox{ and }\alpha\in[0,1].$ (13)

First of all, we prove that $A$ is the indicator function of a point, that is,
$A=\text{I}_{\\{(x,y)\\}}$ for an $(x,y)\in\mathbb{R}^{2}$. By the definition
of support function, we have that

$\operatorname{med}(s_{\mathcal{X}}(u,\alpha))=\operatorname{med}(\\{\langle
u,(0,-1)\rangle,\langle u,(2,0)\rangle,\langle u,(1,1)\rangle\\})$

for every $u\in\mathbb{S}^{1}$ and $\alpha\in[0,1]$. As the median of three
distinct real points is a single point, taking (13) into account, we have that
$s_{A}(u,\alpha)=\operatorname{med}(\\{\langle u,(0,-1)\rangle,\langle
u,(2,0)\rangle,\langle u,(1,1)\rangle\\})$ for every $u\in\mathbb{S}^{1}$ and
$\alpha\in[0,1].$ Thus, $s_{A}(u,\alpha)$ does not change with $\alpha.$
Additionally,
$\operatorname{med}(s_{\mathcal{X}}(-u,\alpha))=-\operatorname{med}(s_{\mathcal{X}}(u,\alpha))$
for every $u\in\mathbb{S}^{1}$ and $\alpha\in[0,1]$. We prove that the
supremum and the infimum of the projection of every $\alpha$-level of $A$ in
every direction $u$ coincides, which implies that $A$ is the indicator
function of a single point of $\mathbb{R}^{2}$. On the one hand

$\operatorname{med}(s_{\mathcal{X}}(u,\alpha))=s_{A}(u,\alpha)=\sup_{v\in
A_{\alpha}}\langle u,v\rangle.$

On the other hand

$\displaystyle-\operatorname{med}(s_{\mathcal{X}}(u,\alpha))=\operatorname{med}(s_{\mathcal{X}}(-u,\alpha))=s_{A}(-u,\alpha)=$
$\displaystyle\sup_{v\in A_{\alpha}}\langle-u,v\rangle=\sup_{v\in
A_{\alpha}}-\langle u,v\rangle=-\inf_{v\in A_{\alpha}}\langle u,v\rangle.$

Thus

$\inf_{v\in A_{\alpha}}\langle
u,v\rangle=\operatorname{med}(s_{\mathcal{X}}(u,\alpha))=\sup_{v\in
A_{\alpha}}\langle u,v\rangle$

for every $u\in\mathbb{S}^{1}$ and $\alpha\in[0,1]$. It implies that
$A=\text{I}_{\\{(x,y)\\}}$ for some $(x,y)\in\mathbb{R}^{2}$.

Now, we prove that there is no such $(x,y)\in\mathbb{R}^{2}$ such that
$A=\text{I}_{\\{(x,y)\\}}$. Let $(1,0)\in\mathbb{S}^{1}$. It is clear that

$x=s_{A}((1,0),\alpha)=\operatorname{med}(\\{0,2,1\\})=1$

for every $\alpha\in[0,1]$, thus $x=1$. If we take $(0,1)\in\mathbb{S}^{1}$,

$y=s_{A}((0,1),\alpha)=\operatorname{med}(\\{-1,0,1\\})=0$

for every $\alpha\in[0,1]$ and $y=1$.

The support median $A$ must be the fuzzy number $\text{I}_{\\{(1,0)\\}}$. But,
if we consider the direction $u=(1/\sqrt{5},-2/\sqrt{5})\in\mathbb{S}^{1}$,

$s_{A}(u,\alpha)=\cfrac{1}{\sqrt{5}}\neq\cfrac{2}{\sqrt{5}}=\operatorname{med}(\\{\cfrac{2}{\sqrt{5}},\cfrac{2}{\sqrt{5}},-\cfrac{1}{\sqrt{5}}\\})$

and $s_{A}(u,\alpha)\neq\operatorname{med}(s_{\mathcal{X}}(u,\alpha))$ for
every $\alpha\in[0,1]$. This implies that the support median for the fuzzy
random variable $\mathcal{X}$ does not exist.

Since the example uses indicator functions of crisp points of
${\mathbb{R}}^{2}$, it indicates that already in the case of (crisp) bivariate
data it is not possible, in general, to define a median by requiring its
projection in every direction to be a median of the corresponding projection
of the random vector. Therefore this behaviour is not a ‘flaw’ of the notion
of a support median for fuzzy data.

The notion of a support median is ad hoc in the sense that its definition is
natural (given the identification between an element of
${\mathcal{F}_{c}}({\mathbb{R}}^{p})$ and its support function) but that does
not prove, by itself, that support medians will inherit the nice theoretical
properties usually ascribed to the median. In the remainder of the paper, we
explore the idea that the framework of statistical depth functions for fuzzy
data and fuzzy random variables can adequately supply mathematical criteria by
which some or all support medians are the unique fuzzy sets satisfying certain
nice properties. To that end, we introduce now the definition of a depth
median.

######  Definition 3.4.

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ and $D$ be a depth
function. We define the set of depth medians (or $D$-medians), of
$\mathcal{X}$ based on $\mathcal{J}\subseteq\mathcal{F}_{c}(\mathbb{R}^{p})$
as

$\operatorname{Med}_{\mathcal{J}}(\mathcal{X};D)=\\{A\in\mathcal{J}\mathrel{\mathop{\mathchar
58\relax}}D(A;\mathcal{X})=\max_{U\in\mathcal{J}}D(U;\mathcal{X})\\},$ (14)

where $D(A;\mathcal{X})$ is the depth function evaluated at $A$ with respect
to the distribution of $\mathcal{X}.$ The elements of
$\operatorname{Med}_{\mathcal{J}}(\mathcal{X};D)$ are called depth medians,
$D$-medians, of $\mathcal{X}$ based on $\mathcal{J}$.

When $\mathcal{J}={\mathcal{F}_{c}}({\mathbb{R}}^{p}),$ as occurs in the rest
of this work, we drop the $\mathcal{J}$ in (14), making use of the notation
$\operatorname{Med}(\mathcal{X};D)$ and refer to it as the set of depth
medians, $D$-medians, of $\mathcal{X}.$

It should be understood that maximizing some depth function does not
automatically imply having the usual properties of a median, such as
robustness. This is exemplified in the multivariate case by zonoid depth [8]
or the depth based on large deviations [37]. In both cases, when being in
dimension one, the maximum is attained at the mean, which is not a robust
location estimator. However it is reasonable to think that maximizing a depth
function is a nice property for a location estimator, specially in view of the
properties that generally constitute the notion of depth (maximization at the
center of symmetry, decreasing depth as one moves away from the center,
vanishing depth when going to infinity, …).

We would like to close this section by recalling that not all available
definitions of a median for fuzzy data are support medians. For instance, [36,
Definition 4.1] uses a mid-ldev-rdev representation. It is based on minimizing
the $L^{1}$-distance for an appropriate metric and thus it is a depth median
for the corresponding $L^{1}$-depth associated to that metric.

##  4 Main results

The results in this section connect the concepts in Section 3. In particular:
$D$-medians ($\operatorname{Med}(\cdot;D)$), support medians
($\operatorname{Med}_{s}(\cdot)$), and $1$-medians
($\operatorname{Med}_{1}(\cdot)$). We start by noting that $1$-medians can
easily be recast as the maximizers of the $L^{1}$-depth, in (2), that makes
use of the $\rho_{1}$-metric.

######  Proposition 4.1.

Let $\mathcal{X}\in L^{1}[{\mathcal{F}_{c}}({\mathbb{R}})].$ Then,
$\operatorname{Med}_{1}(\mathcal{X})=\operatorname{Med}(\mathcal{X};D_{1}).$

The proof easily follows from the fact that minimizing
$E[\rho_{1}(\cdot,\mathcal{X})]$ is equivalent to maximizing
$D_{1}(\cdot;\mathcal{X})=(1+E[\rho_{1}(\cdot,\mathcal{X})])^{-1}$.

We establish next the first important result in this work: $1$-medians, or
$D_{1}$-medians, are exactly the support medians. This result solves the
minimization problem in (9).

######  Theorem 4.2.

Let $\mathcal{X}\in L^{1}[{\mathcal{F}_{c}}({\mathbb{R}})].$ Then,
$\operatorname{Med}_{1}(\mathcal{X})=\operatorname{Med}_{s}(\mathcal{X})$.

The interest of Theorem 4.2 goes both ways. From the perspective of
$1$-medians, it provides a useful characterization by which they can be
obtained without the burden of solving an stochastic optimization problem over
the whole ${\mathcal{F}_{c}}({\mathbb{R}})$. From the perspective of support
medians, it shows that they are the unique fuzzy sets with the
$L^{1}$-minimization property that has been used as a mathematical definition
of the median in a metric space since Fréchet [11, position équiprobable,
p.227].

An application of Theorem 4.2 is in proving that $D_{1}$ satisfies one of the
properties constituting the notion of geometric fuzzy depth [13]. For a depth
function $D,$ based on $\mathcal{H}\subseteq
L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ and
$\mathcal{J}\subseteq\mathcal{F}_{c}(\mathbb{R}^{p}),$ this property, known as
P3b, states that for each $\mathcal{X}\in\mathcal{H}$ and each
$A\in\mathcal{J}$ such that
$D(A;\mathcal{X})=\sup\\{D(B;\mathcal{X})\mathrel{\mathop{\mathchar
58\relax}}B\in\mathcal{J}\\},$ $D$ fulfills

$\displaystyle D(A;\mathcal{X})\geq D(U;\mathcal{X})\geq D(V;\mathcal{X})$

$\mbox{for all }U,V\in\mathcal{J}\mbox{ satisfying }d(A,V)=d(A,U)+d(U,V),$
where $d\mathrel{\mathop{\mathchar
58\relax}}\mathcal{F}_{c}(\mathbb{R}^{p})\times\mathcal{F}_{c}(\mathbb{R}^{p})\rightarrow[0,\infty)$
is a metric. For the fulfillment of other properties, see González-De La
Fuente et al. [15].

######  Theorem 4.3.

Let $\mathcal{X}\in L^{1}[\mathcal{F}_{c}(\mathbb{R})].$ Then, $D_{1}$
satisfies P3b for the $\rho_{1}$ metric.

The proof is based on that Theorem 4.2 characterizes the fuzzy sets that
maximize $D_{1},$ as they are support medians. In [15], property P3b was
studied for the $\rho_{r}$ metrics but results for $\rho_{1}$ could not be
obtained with that method of proof since the $L^{1}$-norm between support
functions is not strictly convex.

We turn now to showing that the support medians are exactly the depth medians
induced by the notion of Tukey depth, in (3). Interestingly, this
characterization of support medians does not require the random variable to be
in $L^{1}[\mathcal{F}_{c}(\mathbb{R})],$ and so it is valid for every fuzzy
random number.

######  Theorem 4.4.

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})].$ Then,
$\operatorname{Med}_{s}(\mathcal{X})=\operatorname{Med}(\mathcal{X};D_{FT})$.

A direct implication of Proposition 4.1 and Theorems 4.2 and 4.4, is the
following corollary.

######  Corollary 4.5.

Let $\mathcal{X}\in L^{1}[{\mathcal{F}_{c}}({\mathbb{R}})].$ Then,
$\operatorname{Med}(\mathcal{X};D_{1})=\operatorname{Med}(\mathcal{X};D_{FT}).$

We turn now to $D_{FP}$-medians, which are based on the outlier function
$O(\cdot;\mathcal{X})$ in (4). In order for that function to be well defined,
it is required for $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ to
be non-degenerate (i.e., it is not identical almost surely to some constant
$A\in{\mathcal{F}_{c}}({\mathbb{R}}^{p})$.) Next result states that
$D_{FP}$-medians are support medians. In fact, the unique $D_{FP}$-median is
$\operatorname{med}_{Si}(\mathcal{X})$.

######  Proposition 4.6.

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})]$ be non-degenerate.
Then,

$\displaystyle\operatorname{Med}(\mathcal{X};D_{FP})=\\{\operatorname{med}_{Si}(\mathcal{X})\\}\subseteq\operatorname{Med}_{s}(\mathcal{X}).$
(15)

Next result (Theorem 4.8) studies when the inclusion in (15) becomes an
equality. For that, the theorem has an assumption of median uniqueness for
$s_{\mathcal{X}}(u,\alpha),$ which cannot be removed as the following example
shows.

######  Example 4.7.

Given the probability space
$(\\{\omega_{1},\omega_{2}\\},\mathcal{P}(\\{\omega_{1},\omega_{2}\\}),\mathbb{P})$,
with $\mathbb{P}(\omega_{1})=\mathbb{P}(\omega_{2})$, let
$\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\\{\omega_{1},\omega_{2}\\}\rightarrow\mathcal{F}_{c}(\mathbb{R})$
be a fuzzy random variable such that
$\mathcal{X}(\omega_{1})=\text{I}_{\\{1\\}}$ and
$\mathcal{X}(\omega_{2})=\text{I}_{\\{3\\}}$.

We have $\operatorname{med}(s_{\mathcal{X}}(1,\alpha))=[1,3]$ and
$\operatorname{med}(s_{\mathcal{X}}(-1,\alpha))=[-3,-1]$ for every
$\alpha\in[0,1]$. Using Theorem 4.2, the set of 1-medians of $\mathcal{X}$ is

$\operatorname{Med}_{1}(\mathcal{X})=\\{U\in\mathcal{F}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}s_{U}(u,\alpha)\in[\min\\{u,3u\\},\max\\{u,3u\\}]\text{ for all
}(u,\alpha)\in\mathbb{S}^{0}\times[0,1]\\}$

Let us consider the set
$A=\text{I}_{\\{2\\}}\in\operatorname{Med}_{1}(\mathcal{X})$. By convention
about the univariate median in the definition of $D_{FP}$, we take
$\operatorname{med}(s_{\mathcal{X}}(u,\alpha))=2u$ and
$\text{MAD}(s_{\mathcal{X}}(u,\alpha))=1$ for all
$(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$, thus

$O(A;\mathcal{X})=\sup_{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]}|s_{A}(u,\alpha)-2u|=0.$

Thus $D_{FP}(A;\mathcal{X})=1$ and $A$ necessarily maximizes the projection
depth. Now let us consider $B=T(1,2,3)\in\operatorname{Med}_{1}(\mathcal{X})$.
We have $B_{\alpha}=[1+\alpha,3-\alpha]$, $s_{B}(1,\alpha)=3-\alpha$ and
$s_{B}(-1,\alpha)=-1-\alpha$ for every $\alpha\in[0,1]$. Thus

$O(B;\mathcal{X})=\sup_{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]}|s_{B}(u,\alpha)-2|=\sup_{\alpha\in[0,1]}|1-\alpha|=1$

Thus $D_{FP}(B;\mathcal{X})=1/2<D_{FP}(A;\mathcal{X})$ but
$A,B\in\operatorname{Med}_{1}(\mathcal{X})=\operatorname{Med}_{s}(\mathcal{X})$.

######  Theorem 4.8.

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})]$ be non-degenerate and
such that the median of $s_{\mathcal{X}}(u,\alpha)$ is unique for each
$u\in\mathbb{S}^{0}$ and $\alpha\in[0,1]$. Then

$\operatorname{Med}_{s}(\mathcal{X})=\operatorname{Med}(\mathcal{X};D_{FP})=\\{\operatorname{med}_{Si}(\mathcal{X})\\}=\\{\operatorname{med}_{Gr}(\mathcal{X})\\}.$

Let us denote by $C^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ the set of all fuzzy
random variables $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ such
that $s_{\mathcal{X}}(u,\alpha)$ is a continuous random variable for each
$(u,\alpha)\in\mathbb{S}^{p-1}\times[0,1]$. As shown in [14], fuzzy
generalizations of simplicial depth have nice properties when computed with
respect to $\mathcal{X}\in C^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})].$ In that
situation, there is a unique support median that maximizes both the modified
simplicial depth and the fuzzy simplicial depth, in (5) and (6) respectively.

######  Theorem 4.9.

Let $\mathcal{X}\in C^{0}[\mathcal{F}_{c}(\mathbb{R})]$. Then

$\operatorname{Med}_{s}(\mathcal{X})=\operatorname{Med}(\mathcal{X};D_{mS})=\operatorname{Med}(\mathcal{X};D_{FS})=\\{\operatorname{med}_{Si}(\mathcal{X})\\}=\\{\operatorname{med}_{Gr}(\mathcal{X})\\}.$

If the fuzzy random variable $\mathcal{X}\not\in
C^{0}[\mathcal{F}_{c}(\mathbb{R})]$, both
$\operatorname{Med}_{s}(\mathcal{X})\neq\operatorname{Med}(\mathcal{X};D_{FS})$
and
$\operatorname{Med}_{s}(\mathcal{X})\neq\operatorname{Med}(\mathcal{X};D_{mS})$
can happen, as shown by the following example.

######  Example 4.10.

Let $X$ be a real random variable with cumulative distribution function

$\mathbb{F}_{X}(x)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{lcr}0,&\text{ if }x\in(-\infty,0)\\\
.145x,&\text{ if }x\in[0,2)\\\ .01x+.47,&\text{ if }x\in[2,53)\\\ 1,&\text{ if
}x\in[53,\infty).\end{array}\right.$

Note that $\operatorname{Med}(X)=\\{3\\},$ which implies, setting
$\mathcal{X}=\text{I}_{\\{X\\}}\in L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$,

$\operatorname{Med}_{s}(\mathcal{X})=\left\\{\text{I}_{\\{3\\}}\right\\}.$
(16)

Let us consider the simplicial depth based on
$L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$ and $\mathcal{F}_{c}(\mathbb{R})$.
Using [14, Proposition 5.1] (reproduced as Lemma 5.5 below),

$D_{FS}\left(\text{I}_{\\{x\\}};\mathcal{X}\right)=1-(1-\mathbb{F}_{X}(x))^{2}-(\mathbb{F}_{X}(x)-\mathbb{P}(X=x))^{2},$

and, therefore,

$D_{FS}(\text{I}_{\\{3\\}};\mathcal{X})=.5\mbox{ and
}D_{FS}(\text{I}_{\\{2\\}};\mathcal{X})=.6558.$

Taking into account (16), that implies
$\operatorname{Med}_{s}(\mathcal{X})\neq\operatorname{Med}(\mathcal{X};D_{FS})$
because $I_{\\{3\\}}$ does not maximize $D_{FS}$. The case of $D_{mS}$ is done
analogously.

######  Remark 4.11.

One of the main uses of depth functions is to generalize the concept of a
median as the element with maximal depth. It follows from the results in this
section that, for $p=1$, the confluence of the notions of 1-median, support
median and depth median for the depth functions $D_{FT}$, $D_{FP}$, $D_{1}$,
$D_{mS}$ and $D_{FS}$, is nice and natural. The fact that support medians may
not exist in the case $p>1$ underlines that depth medians are a reasonable
approach to defining medians for multivariate fuzzy data.

In our study of the case $p=1$, in some cases we have resorted to additional
assumptions which seem to be reasonable. For example, the definition of
projection depth presupposes that the univariate median is taken to be the
midpoint of all possible medians, whereas the definition of a support median
allows for arbitrary medians; studying the relationship between both concepts
under the assumption of a unique median is natural. In the case of simplicial
depths, they are generalizations of multivariate simplicial depth which enjoys
good properties for continuous variables, whence restricting ourselves to
$C^{0}[{\mathcal{F}_{c}}({\mathbb{R}}^{p})]$ is natural.

Next we present two examples in which the set of medians of a fuzzy random
variable is computed explicitly.

######  Example 4.12.

Let
$(\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\},\mathcal{P}(\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}),\mathbb{P})$
be a probabilistic space such that
$\mathbb{P}(\omega_{1})=\mathbb{P}(\omega_{2})=\mathbb{P}(\omega_{3})=\mathbb{P}(\omega_{4})$.
Let $\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{F}_{c}(\mathbb{R})$ be a fuzzy random
variable defined by $\mathcal{X}(\omega_{1})=T(1,2,3)$,
$\mathcal{X}(\omega_{2})=\text{I}_{\\{4\\}}$,
$\mathcal{X}(\omega_{3})=\text{I}_{\\{5\\}}$ and
$\mathcal{X}(\omega_{4})=T(6,6,7)$. By Theorem 4.2, the set of $1$-medians is
the set of support medians:

$\text{Med}_{1}(\mathcal{X})=\\{A\in\mathcal{F}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}s_{A}(u,\alpha)\in\text{Med}(s_{\mathcal{X}}(u,\alpha))\text{ for
every }u\in\mathbb{S}^{0},\alpha\in[0,1]\\}$

The support function of $\mathcal{X}(\omega_{i})$, with $i\in\\{1,2,3,4\\}$,
has the following expressions:

$s_{\mathcal{X}(\omega_{1})}(u,\alpha)=\left\\{\begin{array}[]{lcc}3-\alpha&\text{
if }u=1,\alpha\in[0,1],\\\ \\\ -1-\alpha&\text{ if }u=-1,\alpha\in[0,1]\\\
\end{array}\right.$

$s_{\mathcal{X}(\omega_{2})}(u,\alpha)=4u$

$s_{\mathcal{X}(\omega_{3})}(u,\alpha)=5u$

and

$s_{\mathcal{X}(\omega_{4})}(u,\alpha)=\left\\{\begin{array}[]{lcc}7-\alpha&\text{
if }u=1,\alpha\in[0,1],\\\ \\\ -6&\text{ if }u=-1,\alpha\in[0,1]\\\
\end{array}\right.$

Therefore

$\text{Med}(s_{\mathcal{X}}(1,\alpha))=\text{Med}(\\{3-\alpha,4,5,7-\alpha\\})=[4,5]$

and

$\text{Med}(s_{\mathcal{X}}(-1,\alpha))=\text{Med}(\\{-1-\alpha,-4,-5,-6\\})=[-5,-4]$

Thus, the set of $1$-medians is

$\text{Med}_{1}(\mathcal{X})=\\{M\in\mathcal{F}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}M_{\alpha}\subseteq[4,5]\text{ for all }\alpha\in[0,1]\\}$

By Theorem 4.4, the above set is also the set of maximizers of $D_{FT}$. It is
graphically represented in Figure 1.

Figure 1: Visual representation of the set of fuzzy sets maximizing
$D_{FT}(\cdot;\mathcal{X})$. Every fuzzy set ‘contained’ in the blue area has
maximal depth.

######  Example 4.13.

Let $\Omega=\\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\\}$ and
$(\Omega,\mathcal{P}(\Omega),\mathbb{P})$ be a probabilistic space such that
$\mathbb{P}(\omega_{1})=\mathbb{P}(\omega_{2})=\mathbb{P}(\omega_{3})=\mathbb{P}(\omega_{4})$.
Let $\mathcal{X}\mathrel{\mathop{\mathchar
58\relax}}\Omega\rightarrow\mathcal{F}_{c}(\mathbb{R})$ be a fuzzy random
variable defined by $\mathcal{X}(\omega_{1})=T(1,2,3)$,
$\mathcal{X}(\omega_{2})=T(4,4,5)$, $\mathcal{X}(\omega_{3})=T(6,7,8)$ and
$\mathcal{X}(\omega_{4})=\text{I}_{\\{9\\}}$.

Let us compute the set of support medians. For that, we compute the
$\alpha$-levels for each $\mathcal{X}(\omega_{i}),$ $i=1,\ldots,4,$ obtaining,
for any $\alpha\in[0,1],$
$(\mathcal{X}(\omega_{1}))_{\alpha}=[\alpha+1,3-\alpha],$
$(\mathcal{X}(\omega_{2}))_{\alpha}=[4,5-\alpha],$
$(\mathcal{X}(\omega_{3}))_{\alpha}=[\alpha+6,8-\alpha]\mbox{ and
}(\mathcal{X}(\omega_{4}))_{\alpha}=\\{9\\}.$ Then the support functions of
those fuzzy sets are

$s_{\mathcal{X}(\omega_{1})}(u,\alpha)=\left\\{\begin{array}[]{lcc}3-\alpha&\text{
if }u=1,\alpha\in[0,1],\\\ -1-\alpha&\text{ if
}u=-1,\alpha\in[0,1]\end{array}\right.$
$s_{\mathcal{X}(\omega_{2})}(u,\alpha)=\left\\{\begin{array}[]{lcc}5-\alpha&\text{
if }u=1,\alpha\in[0,1],\\\ -4&\text{ if
}u=-1,\alpha\in[0,1]\end{array}\right.$
$s_{\mathcal{X}(\omega_{3})}(u,\alpha)=\left\\{\begin{array}[]{lcc}8-\alpha&\text{
if }u=1,\alpha\in[0,1],\\\ -6-\alpha&\text{ if
}u=-1,\alpha\in[0,1]\end{array}\right.$

and $s_{\mathcal{X}(\omega_{4})}(u,\alpha)=9u.$ Consequently, for all
$\alpha\in[0,1],$

$\text{Med}(s_{\mathcal{X}}(1,\alpha))=\text{Med}(\\{3-\alpha,5-\alpha,8-\alpha,9\\})=[5-\alpha,8-\alpha]$

and

$\text{Med}(s_{\mathcal{X}}(-1,\alpha))=\text{Med}(\\{-1-\alpha,-4,-6-\alpha,-9\\})=[-4,-6-\alpha].$

Thus, the set of support medians, which coincides with the 1-medians (Theorem
4.2) and the Tukey medians (Theorem 4.4) is

$\operatorname{Med}_{1}(\mathcal{X})=\\{A\in\mathcal{F}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\inf A_{\alpha}\in[4,6+\alpha],\sup
A_{\alpha}\in[5-\alpha,8-\alpha]\text{ for all }\alpha\in[0,1]\\},$

displayed in Figure 2.

Figure 2: Illustration (in blue and red) of the set of medians of
$\mathcal{X}$. The areas where the left and right slope of a median must be
contained are represented in different colors.

##  5 Proofs

###### Proof of Proposition 3.2.

Let $\mathcal{X}$ be a fuzzy random variable in ${\mathbb{R}}$ and let
$A\in\operatorname{Med}_{s}(\mathcal{X})$ be a support median of
$\mathcal{X}$. The $\alpha$-level of $A$ can be expressed in terms of its
support function as

$A_{\alpha}=[-s_{A}(-1,\alpha),s_{A}(1,\alpha)]=[\inf A_{\alpha},\sup
A_{\alpha}]$

whenever $\alpha\in[0,1]$. By definition
$s_{A}(u,\alpha)\in\operatorname{Med}(s_{\mathcal{X}}(u,\alpha))$ whenever
$u\in\mathbb{S}^{0},\alpha\in[0,1]$. Thus

$\displaystyle-s_{A}(-1,\alpha)$
$\displaystyle\in\operatorname{Med}(-s_{\mathcal{X}}(-1,\alpha))=\operatorname{Med}(\inf\mathcal{X}_{\alpha})$
$\displaystyle s_{A}(1,\alpha)$
$\displaystyle\in\operatorname{Med}(s_{\mathcal{X}}(1,\alpha))=\operatorname{Med}(\sup\mathcal{X}_{\alpha})$

Let $\operatorname{med}_{Gr}(\mathcal{X})$ be the fuzzy median of
$\mathcal{X}$ in (7). By (8),

$\underline{\operatorname{med}}(\inf\mathcal{X}_{\alpha})\leq-
s_{A}(-1,\alpha)\mbox{ and
}s_{A}(1,\alpha)\leq\overline{\operatorname{med}}(\sup\mathcal{X}_{\alpha}).$

Thus $A_{\alpha}\subseteq(\operatorname{med}_{Gr}(\mathcal{X}))_{\alpha}$ for
every $\alpha\in[0,1]$. Consequently, the support median $A$ is contained in
$\operatorname{med}_{Gr}(\mathcal{X})$. ∎

The next characterization is used for the proof of Theorem 4.2.

######  Lemma 5.1 ([5]).

Let $X$ be a real random variable and $x\in\mathbb{R}$. Then,
$x\in\operatorname{Med}(X)$ if and only if

$\text{E}[|X-y|]=\inf_{z\in\mathbb{R}}\text{E}[|X-z|]$

###### Proof of Theorem 4.2.

Let $\mathcal{X}$ be a fuzzy random variable. The proof of the result in (10)
([35, Theorem 4.1]) already shows that
$\operatorname{Med}_{s}(\mathcal{X})\subseteq\operatorname{Med}_{1}(\mathcal{X})$
for every fuzzy random variable $\mathcal{X}$. That is so because the proof
that $\operatorname{med}_{Si}(\mathcal{X})$ is a $1$-median only uses the fact
that the endpoints of each $\alpha$-cut are medians. Notice that
$\inf\mathcal{X}_{\alpha}=-s_{\mathcal{X}}(-1,\alpha)$ and
$\sup\mathcal{X}_{\alpha}=s_{\mathcal{X}}(1,\alpha)$, so reasoning with the
endpoints is equivalent to reasoning with the support function.

There remains to prove
$\operatorname{Med}_{1}(\mathcal{X})\subseteq\operatorname{Med}_{s}(\mathcal{X})$.
Reasoning by contradiction, assume there exists
$B\in\operatorname{Med}_{1}(\mathcal{X})\backslash\operatorname{Med}_{s}(\mathcal{X})$.
Then the set

$K=\\{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}s_{B}(u,\alpha)\not\in\operatorname{Med}(s_{\mathcal{X}}(u,\alpha))\\}$

is not empty. We consider the measure $\upsilon\otimes\nu$ over the set
$\mathbb{S}^{0}\times[0,1]$, where $\upsilon$ denotes the uniform distribution
over $\mathbb{S}^{0}$ and $\nu$ denotes the Lebesgue measure over $[0,1]$.

Fix any $A\in\operatorname{Med}_{s}(\mathcal{X})$. Clearly,

$\text{E}[\rho_{1}(\mathcal{X},B)]=\text{E}[\rho_{1}(\mathcal{X},A)]$

Using Fubini’s theorem, which is possible due to the joint measurability of
$s_{\mathcal{X}(\cdot)}(\cdot,\cdot)$ in its three arguments [1, Proposition
4.4],

$\int_{\mathbb{S}^{0}}\int_{[0,1]}\text{E}[|s_{\mathcal{X}}(u,\alpha)-s_{B}(u,\alpha)|]dud\alpha=\int_{\mathbb{S}^{0}}\int_{[0,1]}\text{E}[|s_{\mathcal{X}}(u,\alpha)-s_{A}(u,\alpha)|]dud\alpha$
(17)

By Lemma 5.1, if $x\in\mathbb{R}$ is not a median of a real random variable
$X$, then $\text{E}[|X-x|]>\inf_{y}\text{E}[|X-y|]$. Using (17), we deduce
that $K$ has $(\upsilon\otimes\nu)$-measure $0$, that is
$(\upsilon\otimes\nu)(K)=0$. Set $H=\mathbb{S}^{0}\times[0,1]\setminus K$.
Thus $H$ has $(\upsilon\otimes\nu)$-measure $1$ and $H$ is a dense subset of
$\mathbb{S}^{0}\times[0,1]$. As $\mathbb{S}^{0}=\\{-1,1\\}$, we can express
$H$ as the union of $\\{1\\}\times U_{1}$ and $\\{-1\\}\times U_{-1}$, where
$U_{1}$ and $U_{-1}$ are dense subsets of $[0,1]$.

Since $K$ is non-empty, there is some $(u,\alpha)\in K$. Without loss of
generality, we assume $u=1$ (if $u=-1$ the argument is analogous or,
alternatively, the same argument applies to $-\mathcal{X}$). We distinguish
two cases, at the end of each we will reach a contradiction.

Case I. ($\alpha>0$) As $U_{1}$ is dense in $[0,1]$, there exists a non-
decreasing sequence $\\{\alpha_{n}\\}_{n}$ of elements of $U_{1}$ such that
$(1,\alpha_{n})\in H$ for all $n\in\mathbb{N}$ and
$\lim_{n}\alpha_{n}=\alpha$. As $\\{\alpha_{n}\\}_{n}$ is non-decreasing,
$\\{s_{B}(1,\alpha_{n})\\}_{n}$ is a non-increasing sequence with
$s_{B}(1,\alpha_{n})\geq s_{B}(1,\alpha)$ for all $n\in\mathbb{N}$. By [25],
$s_{U}(\cdot,\alpha)$ is left-continuous for all $\alpha\in[0,1]$ and for all
$U\in\mathcal{F}_{c}(\mathbb{R})$. Thus
$\lim_{n}s_{B}(1,\alpha_{n})=s_{B}(1,\alpha)$. Analogously,
$\lim_{n}s_{\mathcal{X}(\omega)}(1,\alpha_{n})=s_{\mathcal{X}(\omega)}(1,\alpha)$
for each $\omega\in\Omega$.

First,

$\displaystyle\mathbb{P}(s_{\mathcal{X}}(1,\alpha)<s_{B}(1,\alpha))\leq\lim\inf_{n}\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})<s_{B}(1,\alpha))\leq$
$\displaystyle\lim\inf_{n}\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})<s_{B}(1,\alpha_{n}))=\lim\inf_{n}\left(1-\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})\geq
s_{B}(1,\alpha_{n}))\right)\leq\cfrac{1}{2},$

where the second inequality is due to $s_{B}(1,\alpha)\leq
s_{B}(1,\alpha_{n})$ for all $n\in\mathbb{N}$ and third inequality to
$(1,\alpha_{n})\in H$, thus
$s_{B}(1,\alpha_{n})\in\operatorname{Med}(s_{\mathcal{X}}(1,\alpha_{n}))$ for
all $n\in\mathbb{N}$. Hence $\mathbb{P}(s_{\mathcal{X}}(1,\alpha)\geq
s_{B}(1,\alpha))\geq 1/2$.

Denoting by $F_{u,\alpha}$ the cumulative distribution function of the real
random variable $s_{\mathcal{X}}(u,\alpha)$,

$\cfrac{1}{2}\leq\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})\leq
s_{B}(1,\alpha_{n}))\leq\mathbb{P}(s_{\mathcal{X}}(1,\alpha)\leq
s_{B}(1,\alpha_{n}))=F_{1,\alpha}(s_{B}(1,\alpha_{n})),$

where the first inequality is due to $(1,\alpha_{n})\in H$ and second one
holds because $s_{\mathcal{X}(\omega)}(1,\alpha_{n})\geq
s_{\mathcal{X}(\omega)}(1,\alpha)$ for all $n\in\mathbb{N}$ and
$\omega\in\Omega$. Taking limits at both sides,

$\cfrac{1}{2}\leq\lim_{n\rightarrow\infty}F_{1,\alpha}(s_{B}(1,\alpha_{n}))=F_{1,\alpha}(s_{B}(1,\alpha)),$

where the equality is due to the continuity from the right of the cumulative
distribution function. Thus
$s_{B}(1,\alpha)\in\operatorname{Med}(s_{\mathcal{X}}(1,\alpha))$ but
$(1,\alpha)\in K$, a contradiction.

Case II.($\alpha=0$) As $U_{1}$ is dense in $[0,1]$, there exists a non-
increasing sequence $\\{\alpha_{n}\\}_{n}$ such that $(1,\alpha_{n})\in H$ for
all $n\in\mathbb{N}$ and $\lim_{n}\alpha_{n}=0$. Then, the sequence
$\\{s_{U}(\alpha_{n})\\}_{n}$ is non-decreasing and such that
$s_{U}(1,\alpha_{n})\leq s_{U}(1,0)$ for all $n\in\mathbb{N}$ and for all
$U\in\mathcal{F}_{c}(\mathbb{R})$. By [25], the function $s_{U}(\cdot,\alpha)$
is right-continuous at $\alpha=0$, thus

$\lim_{n\rightarrow\infty}s_{U}(1,\alpha_{n})=s_{U}(1,0),$

for all $U\in\mathcal{F}_{c}(\mathbb{R})$.

We prove that $s_{B}(1,0)\in\operatorname{Med}(s_{\mathcal{X}}(1,0))$. First,

$\displaystyle\mathbb{P}(s_{\mathcal{X}}(1,0)>s_{B}(1,0))\leq\lim\inf_{n}\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})>s_{B}(1,0))\leq$
$\displaystyle\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})>s_{B}(1,\alpha_{n}))=\lim\inf_{n}\left(1-\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n}))\right)\leq
s_{B}(1,\alpha_{n}))\leq\cfrac{1}{2},$

where the second inequality is due to $s_{B}(1,\alpha_{n})\leq s_{B}(1,0)$ for
all $n\in\mathbb{N}$ and the third inequality to the fact that
$(1,\alpha_{n})\in H$. This implies that $\mathbb{P}(s_{\mathcal{X}}(1,0)\leq
s_{B}(1,0))\geq 1/2$.

On the other hand,

$\displaystyle\cfrac{1}{2}\leq\mathbb{P}(s_{\mathcal{X}}(1,\alpha_{n})\geq
s_{B}(1,\alpha_{n}))\leq\mathbb{P}(s_{\mathcal{X}}(1,0)\geq
s_{B}(1,\alpha_{n}))=$ $\displaystyle\mathbb{P}(s_{\mathcal{X}}(1,0)\geq
s_{B}(1,0))+\mathbb{P}(s_{\mathcal{X}}(1,0)\in(s_{B}(1,\alpha_{n}),s_{B}(1,0)]),$

where the first inequality is due to $(1,\alpha_{n})\in H$ and the second
holds because $s_{U}(1,\alpha_{n})\leq s_{U}(1,0)$ for all
$U\in\mathcal{F}_{c}(\mathbb{R})$ and for all $n\in\mathbb{N}$. If we take
limits, it is clear that

$\lim_{n\rightarrow\infty}\mathbb{P}(s_{\mathcal{X}}(1,0)\in(s_{B}(1,\alpha_{n}),s_{B}(1,0)])=0,$

because $\lim_{n}s_{B}(1,\alpha_{n})=s_{B}(1,0)$. Thus
$\mathbb{P}(s_{\mathcal{X}}(1,0)\geq s_{B}(1,0))\geq 1/2$ and
$s_{B}(1,0)\in\operatorname{Med}(s_{\mathcal{X}}(1,0))$. That implies
$(1,0)\not\in K$, a contradiction.

We conclude from this that
$\operatorname{Med}_{1}(\mathcal{X})\backslash\operatorname{Med}_{s}(\mathcal{X})$
is empty, which completes the proof. ∎

The following property, which relies on the fact that a convex combination of
$x,y\in{\mathbb{R}}$ is always between $x$ and $y$, is used in the proof of
Theorem 4.3.

######  Lemma 5.2.

Let $x,y\in\mathbb{R}$ and $\lambda\in[0,1]$. Then

$\cfrac{1}{1+\lambda}\cdot(x+\lambda\cdot y)\in[\min\\{x,y\\},\max\\{x,y\\}].$

###### Proof of Theorem 4.3.

Let $A,B,C\in\mathcal{F}_{c}(\mathbb{R})$ be fuzzy sets such that

$\rho_{1}(A,B)=\rho_{1}(A,C)+\rho_{1}(B,C)$ (18)

and $A$ maximizes $D_{1}$. By (10), such $A$ indeed exists and using Theorem
4.2, $s_{A}(u,\alpha)\in\text{Med}(s_{\mathcal{X}}(u,\alpha))$ for every
$u\in\mathbb{S}^{0}$ and $\alpha\in[0,1]$. Thus

$s_{A}(u,\alpha)\in\arg\min_{y\in\mathbb{R}}\text{E}[|y-s_{\mathcal{X}}(u,\alpha)|]$
(19)

for all $u\in\mathbb{S}^{0}$ and $\alpha\in[0,1]$. By (18),

$\displaystyle\int_{\mathbb{S}^{p-1}}\int_{[0,1]}|s_{A}(u,\alpha)-s_{B}(u,\alpha)|dud\alpha=$
$\displaystyle\int_{\mathbb{S}^{p-1}}\int_{[0,1]}(|s_{A}(u,\alpha)-s_{C}(u,\alpha)|+|s_{B}(u,\alpha)-s_{C}(u,\alpha)|)dud\alpha.$

Thus

$\int_{\mathbb{S}^{p-1}}\int_{[0,1]}(|s_{A}(u,\alpha)-s_{C}(u,\alpha)|+|s_{B}(u,\alpha)-s_{C}(u,\alpha)|-|s_{A}(u,\alpha)-s_{B}(u,\alpha)|)dud\alpha=0$

The integrand in the last expression is non-negative due to the triangle
inequality. Thus, there exists a dense subset
$D\subseteq\mathbb{S}^{p-1}\times[0,1]$ with $(\upsilon\otimes\nu)$-measure
$1$ such that

$|s_{A}(u,\alpha)-s_{B}(u,\alpha)|=|s_{A}(u,\alpha)-s_{C}(u,\alpha)|+|s_{B}(u,\alpha)-s_{C}(u,\alpha)|$

for every $(u,\alpha)\in D$. By the necessary conditions for the triangle
inequality to be an equality, for every $(u,\alpha)\in D$ there exists
$\lambda_{u,\alpha}\in[0,1]$ such that

$s_{A}(u,\alpha)-s_{C}(u,\alpha)=\lambda_{u,\alpha}\cdot(s_{C}(u,\alpha)+s_{B}(u,\alpha))$

After some algebra,

$s_{C}(u,\alpha)=\cfrac{1}{1+\lambda_{u,\alpha}}\cdot(s_{A}(u,\alpha)+\lambda_{u,\alpha}\cdot
s_{B}(u,\alpha))$ (20)

Setting $r(u,\alpha)=1/(1+\lambda_{u,\alpha})$,

$\displaystyle\text{E}[\rho_{1}(C,\mathcal{X})]=\int_{\mathbb{S}^{0}\times[0,1]}\text{E}[|s_{C}(u,\alpha)-s_{\mathcal{X}}(u,\alpha)|]dud\alpha=$
$\displaystyle\int_{\mathbb{S}^{0}\times[0,1]}\text{E}[|r(u,\alpha)\cdot
s_{A}(u,\alpha)+(1-r(u,\alpha))\cdot
s_{B}(u,\alpha)-s_{\mathcal{X}}(u,\alpha)|]dud\alpha=$
$\displaystyle\int_{\mathbb{S}^{0}\times[0,1]}\text{E}[|r(u,\alpha)\cdot(s_{A}(u,\alpha)-s_{\mathcal{X}}(u,\alpha))+(1-r(u,\alpha))\cdot(s_{B}(u,\alpha)-s_{\mathcal{X}}(u,\alpha))|]dud\alpha\leq$
$\displaystyle\int_{\mathbb{S}^{0}\times[0,1]}r(u,\alpha)\cdot\text{E}[|s_{A}(u,\alpha)-s_{\mathcal{X}}(u,\alpha)|]+(1-r(u,\alpha))\cdot\text{E}[|s_{B}(u,\alpha)-s_{\mathcal{X}}(u,\alpha)|]dud\alpha\leq$
$\displaystyle\int_{\mathbb{S}^{0}\times[0,1]}\text{E}[|s_{B}(u,\alpha)-s_{\mathcal{X}}(u,\alpha)|]dud\alpha=\text{E}[\rho_{1}(B,\mathcal{X})]$

Since the function $s_{\mathcal{X}}$ is jointly measurable in its three
arguments [20, Lemma 4] the first identity is due to Fubini’s theorem and 1.
The second one is due to (20), the second inequality to the triangle
inequality, and the last identity to (19). Thus $D_{1}(C;\mathcal{X})\leq
D_{1}(B;\mathcal{X})$. ∎

The next result is known and characterizes the set of medians of a real random
variable as the numbers which maximize Tukey depth. It is very useful for the
proof of Theorem 4.4.

######  Lemma 5.3.

Let $X$ be a real random variable. Then, $x\in\operatorname{med}(X)$ if, and
only if, $x\in\arg\max_{y\in\mathbb{R}}HD(y;X)$.

###### Proof.

Let $X$ be a real random variable, and $x\in\mathbb{R}$ such that
$x\in\arg\max_{y\in\mathbb{R}}HD(y;X)$. We assume by contradiction that
$x\not\in\operatorname{Med}(X)$. Without loss of generality, we suppose that
$\mathbb{P}(X\leq x)<1/2$. Let $z\in\operatorname{Med}(X)$. Then

$HD(x;X)=\min\\{\mathbb{P}(X\leq x),\mathbb{P}(X\geq x)\\}<\cfrac{1}{2}\leq
HD(z;X),$ (21)

which is a contradiction.

Now let $x\in\operatorname{Med}(X)$. We must prove that
$x\in\arg\max_{y\in\mathbb{R}}HD(y;\mathbb{R})$. Consider
$x_{0}=\inf\\{y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(X\leq y)\geq 1/2\\}$ and
$x_{1}=\inf\\{y\in\mathbb{R}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{P}(X\geq y)\geq 1/2\\}$. By definition, $\mathbb{P}(X\leq
x_{0})\geq 1/2$ and $\mathbb{P}(X\geq x_{1})\geq 1/2$. The interval
$[x_{0},x_{1}]$ is the set of medians of $X$. If $x_{0}=x_{1}$, the median of
$X$ is unique and the proof is done. If $x_{0}<x_{1}$, the median is not
unique, but $\mathbb{P}(X\in(x_{0},x_{1}))=0$ and every median has the same
halfspace depth. Hence, $x\in\arg\max_{y\in\mathbb{R}}HD(y;X)$. ∎

###### Proof of Theorem 4.4.

Let $A\in\operatorname{Med}_{s}(\mathcal{X})$, i.e.,
$s_{A}(u,\alpha)\in\operatorname{Med}(s_{\mathcal{X}}(u,\alpha))$ for all
$u\in\mathbb{S}^{0}$ and $\alpha\in[0,1]$. Let us show
$A\in\operatorname{Med}(\mathcal{X},D_{FT})$, that is, $A$ maximizes
$D_{FT}(\cdot;\mathcal{X})$. By González-De La Fuente et al. [12], we can
express $D_{FT}(A;\mathcal{X})$ as

$D_{FT}(A;\mathcal{X})=\inf_{u,\alpha}HD(s_{A}(u,\alpha),s_{\mathcal{X}}(u,\alpha)),$

where $HD$ is the Tukey depth for the multivariate case ($\mathbb{R}$ in this
case). For any arbitrary $B\in\mathcal{F}_{c}(\mathbb{R})$,

$\displaystyle D_{FT}(A;\mathcal{X})$
$\displaystyle=\inf_{u,\alpha}HD(s_{A}(u,\alpha),s_{\mathcal{X}}(u,\alpha))=\inf_{u,\alpha}\sup_{y\in\mathbb{R}}HD(y;s_{\mathcal{X}}(u,\alpha))\geq$
$\displaystyle\sup_{y\in\mathbb{R}}\inf_{u,\alpha}HD(y;s_{\mathcal{X}}(u,\alpha))\geq\inf_{u,\alpha}HD(s_{B}(u,\alpha);s_{\mathcal{X}}(u,\alpha))=D_{FT}(B;\mathcal{X}),$

where the second identity uses Lemma 5.3 and the fact that
$A\in\operatorname{Med}_{s}(\mathcal{X})$. That proves
$\mathcal{M}_{\mathcal{X}}\subset\operatorname{Med}(\mathcal{X};D_{FT})$.

For the converse inclusion, let $A\in\operatorname{Med}(\mathcal{X};D_{FT})$.
Reasoning by contradiction, assume there exist
$u\in\mathbb{S}^{0},\alpha\in[0,1]$ such that
$s_{A}(u,\alpha)\not\in\operatorname{Med}(s_{\mathcal{X}}(u,\alpha))$. Without
loss of generality, assume

$\mathbb{P}(s_{\mathcal{X}}(u,\alpha)\leq s_{A}(u,\alpha))<1/2.$

From the definition of Tukey depth for fuzzy sets,
$D_{FT}(A;\mathcal{X})<1/2$. By (10), there exists
$\operatorname{med}_{Si}(\mathcal{X})\in\operatorname{Med}_{1}(\mathcal{X})$
such that
$s_{\operatorname{med}_{Si}}(u,\alpha)\in\operatorname{Med}(s_{\mathcal{X}}(u,\alpha))$
whenever $u\in\mathbb{S}^{0}$, $\alpha\in[0,1]$. Using the first part of the
proof, $D_{FT}(\operatorname{med}_{Si}(\mathcal{X});\mathcal{X})\geq 1/2$.
Thus

$D_{FT}(A;\mathcal{X})<1/2\leq
D_{FT}(\operatorname{med}_{Si}(\mathcal{X});\mathcal{X}),$

yields to a contradiction to the fact that $A$ maximizes
$D_{FT}(\cdot;\mathcal{X})$. ∎

###### Proof of Theorem 4.8.

By the assumption, the support median is unique and so

$\operatorname{Med}_{s}(\mathcal{X})=\\{\operatorname{med}_{Si}(\mathcal{X})\\}=\\{\operatorname{med}_{Gr}(\mathcal{X})\\}.$

Since $\mathcal{X}$ is non-degenerate,
$\text{MAD}(s_{\mathcal{X}}(u,\alpha))>0$ whenever $u\in\mathbb{S}^{0}$,
$\alpha\in[0,1]$. Thus

$O(\operatorname{med}_{Si}(\mathcal{X});\mathcal{X})=0$

As $O(U;\mathcal{X})\geq 0$ for all $U\in\mathcal{F}_{c}(\mathbb{R})$,

$1=D_{FP}(\operatorname{med}_{Si}(\mathcal{X});\mathcal{X})=\sup_{U\in\mathcal{F}_{c}(\mathbb{R})}D_{FP}(U;\mathcal{X}).$

That proves
$\\{\operatorname{med}_{Si}(\mathcal{X})\\}\subseteq\operatorname{Med}(\mathcal{X};D_{FP})$.

Now let $B\in\mathcal{F}_{c}(\mathbb{R})$ such that
$B\neq\operatorname{med}_{Si}(\mathcal{X})$. It implies that there exists
$u\in\mathbb{S}^{0}$ and $\alpha\in[0,1]$ such that $s_{B}(u,\alpha)\neq
s_{\operatorname{med}_{Si}(\mathcal{X})}(u,\alpha)=\operatorname{med}(s_{\mathcal{X}}(u,\alpha))$.
Thus $|s_{B}(u,\alpha)-\operatorname{med}(s_{\mathcal{X}}(u,\alpha))|>0$ and
$O(B;\mathcal{X})>0$. It implies that
$D_{FP}(B;\mathcal{X})<D_{FP}(\operatorname{med}_{Si}(\mathcal{X});\mathcal{X})=1$
and
$\operatorname{Med}(\mathcal{X};D_{FP})=\operatorname{Med}_{s}(\mathcal{X})=\\{\operatorname{med}_{Si}(\mathcal{X})\\}=\\{\operatorname{med}_{Gr}(\mathcal{X})\\}$.
∎

The following is a basic fact about fuzzy numbers.

######  Lemma 5.4.

Let $A,B\in\mathcal{F}_{c}(\mathbb{R})$ and set

$C_{A,B}=\\{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}s_{A}(u,\alpha)\neq s_{B}(u,\alpha)\\}.$

If $A\neq B$, then $(\upsilon\otimes\nu)(C_{A,B})>0$, where $\upsilon$ denotes
the uniform distribution measure over $\mathbb{S}^{0}$ and $\nu$ denotes the
Lebesgue measure over $[0,1]$.

###### Proof.

Let $A,B\in\mathcal{F}_{c}(\mathbb{R})$ be two distinct fuzzy sets, thus
$C_{A,B}\neq\emptyset$. Reasoning by contradiction, assume
$(\upsilon\otimes\nu)(P)=0$. Let $S=\mathbb{S}^{0}\times[0,1]\setminus
C_{A,B}$ be the set of all $(u,\alpha)$ where
$s_{A}(u,\alpha)=s_{B}(u,\alpha)$. Since
$(\upsilon\otimes\nu)(S)=(\upsilon\otimes\nu)(\mathbb{S}^{0}\times[0,1])$, the
set $S$ is dense in $\mathbb{S}^{0}\times[0,1]$. Set
$S_{1}=(\\{1\\}\times[0,1])\cap S$ and $S_{-1}=(\\{-1\\}\times[0,1])\cap S$.
Clearly, $S_{1}$ is dense in $\\{1\\}\times[0,1]$ and $S_{-1}$ is dense in
$\\{-1\\}\times[0,1]$.

Since $C_{A,B}$ is non-empty, let $(u,\alpha)\in C_{A,B}$. We will prove the
case $u=1$, the other one being analogous. We distinguish two cases.

Case I. $(\alpha>0)$. As $S_{1}$ is dense in $\\{1\\}\times[0,1]$, there
exists a monotone increasing sequence $\\{\alpha_{n}\\}_{n}$ such that
$\lim_{n}\alpha_{n}=\alpha$ and $(1,\alpha_{n})\in S$ for all
$n\in\mathbb{N}$. The support function is left-continuous with respect to
$\alpha$ (see [25]), whence

$s_{B}(1,\alpha)=\lim_{n\rightarrow\infty}s_{B}(1,\alpha_{n})=\lim_{n\rightarrow\infty}s_{A}(1,\alpha_{n})=s_{A}(1,\alpha),$

where the second equality is due to $(1,\alpha_{n})\in S$ for every
$n\in\mathbb{N}$. Thus $(1,\alpha)\in S$, which is a contradiction.

Case II. $(\alpha=0)$. In this case, there exists a monotone decreasing
sequence $\\{\alpha_{n}\\}_{n}$ such that $\lim_{n}\alpha_{n}=0$ and
$(1,\alpha_{n})\in S$ for all $n\in\mathbb{N}$. The support function of a
fuzzy set is continuous at $\alpha=0$ (see [25]), thus

$s_{B}(1,0)=\lim_{n\rightarrow\infty}s_{B}(1,\alpha_{n})=\lim_{n\rightarrow\infty}s_{A}(1,\alpha_{n})=s_{A}(1,0),$

like in Case I, thereby reaching a contradiction. ∎

The next result is proved in [14] and it is used in the proof of Theorem 4.9.

######  Lemma 5.5.

([14, Proposition 5.1]) Let $\mathcal{X}\in
L^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$, $U\in\mathcal{F}_{c}(\mathbb{R}^{p})$
and $(u,\alpha)\in\mathbb{S}^{p-1}\times[0,1]$.

$\displaystyle\mathbb{P}(s_{U}(u,\alpha)\in[m(u,\alpha),M(u,\alpha$
$\displaystyle)])=1-[1-F_{u,\alpha}(s_{U}(u,\alpha))]^{p+1}$
$\displaystyle-[F_{u,\alpha}(s_{U}(u,\alpha))-\mathbb{P}(s_{\mathcal{X}}(u,\alpha)=s_{U}(u,\alpha))]^{p+1}.$

If, additionally, $\mathcal{X}\in C^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})],$

$\mathbb{P}(s_{U}(u,\alpha)\in[m(u,\alpha),M(u,\alpha)])=1-[1-F_{u,\alpha}(s_{U}(u,\alpha))]^{p+1}-[F_{u,\alpha}(s_{U}(u,\alpha))]^{p+1}.$

###### Proof of Theorem 4.9.

Let $\mathcal{X}\in C^{0}[\mathcal{F}_{c}(\mathbb{R})]$. In [14] we showed
that, for any $U\in\mathcal{F}_{c}(\mathbb{R})$, $D_{mS}$ and $D_{FS}$ can be
expressed as

$D_{mS}(U;\mathcal{X})=\int_{\mathbb{S}^{0}}\int_{[0,1]}\mathbb{P}[s_{U}(u,\alpha)\in[m(u,\alpha),M(u,\alpha)]]d\nu(\alpha)d\mathcal{V}_{p}(u),$

and

$D_{FS}(U;\mathcal{X})=\inf_{u\in\mathbb{S}^{0}}\int_{[0,1]}\mathbb{P}[s_{U}(u,\alpha)\in[m(u,\alpha),M(u,\alpha)]]d\nu(\alpha).$

Thus, by Lemma 5.5 and the fact that $s_{\mathcal{X}}(u,\alpha)$ is a
continuous real random variable for each
$(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$,

$D_{mS}(U;\mathcal{X})=\int_{\mathbb{S}^{0}}\int_{[0,1]}2F_{u,\alpha}(s_{U}(u,\alpha))[1-F_{u,\alpha}(s_{U}(u,\alpha))]d\nu(\alpha)d\mathcal{V}_{p}(u),$
(22)

and

$D_{FS}(U;\mathcal{X})=\inf_{u\in\mathbb{S}^{0}}\int_{[0,1]}2F_{u,\alpha}(s_{U}(u,\alpha))[1-F_{u,\alpha}(s_{U}(u,\alpha))]d\nu(\alpha).$
(23)

Now, we prove
$\operatorname{Med}_{s}(\mathcal{X})\subseteq\operatorname{Med}(\mathcal{X};D_{mS})$.
Let $A$ be a support median. By (10) and the definition of support median,
$\operatorname{Med}_{s}(\mathcal{X})\neq\emptyset$ and there exists at least
one support median. Because $s_{\mathcal{X}}(u,\alpha)$ is continuous,
$F_{u,\alpha}(s_{A}(u,\alpha))=1/2$ for all
$(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$. Thus, $s_{A}(u,\alpha)$ maximizes
the integrand in (22) for every $(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$. Then
$A\in\operatorname{Med}(\mathcal{X};D_{mS})$. The proof for $D_{FS}$ is
analogous.

We prove that
$\operatorname{Med}(\mathcal{X};D_{mS})\subseteq\operatorname{Med}_{s}(\mathcal{X})$,
since
$\operatorname{Med}_{1}(\mathcal{X})=\operatorname{Med}_{s}(\mathcal{X})$ (see
Theorem 4.2), it is equivalent to prove
$\operatorname{Med}(\mathcal{X};D_{mS})\subseteq\operatorname{Med}_{1}(\mathcal{X})$.
Assume, reasoning by contradiction, that there exists
$V\in\operatorname{Med}(\mathcal{X};D_{mS})$ such that
$V\not\in\operatorname{Med}_{1}(\mathcal{X})$. Then

$\text{E}[\rho_{1}(V;\mathcal{X})]>\inf_{U\in\mathcal{F}_{c}(\mathbb{R})}\text{E}[\rho_{1}(U;\mathcal{X})]=\text{E}[\rho_{1}(A;\mathcal{X})]$
(24)

The set $K=\\{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}s_{V}(u,\alpha)\not\in Med(s_{\mathcal{X}}(u,\alpha))\\}$ can be
expressed as
$K=\\{(u,\alpha)\in\mathbb{S}^{0}\times[0,1]\mathrel{\mathop{\mathchar
58\relax}}F_{u,\alpha}(s_{V}(u,\alpha))\neq 1/2\\}$. In [1, Proposition 4.6]
it is proved that the support function of a fuzzy random variable is jointly
measurable, thus $K$ is a Lebesgue measurable set. From (24),
$(\upsilon\otimes\nu)(K)>0$ (see the proof of Theorem 4.2). The integrand in
(22) is maximized by the medians of $s_{\mathcal{X}}(u,\alpha)$, for each
$(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$. As $K$ has positive measure,

$D_{mS}(V;\mathcal{X})<D_{mS}(A;\mathcal{X})=\sup_{U\in\mathcal{F}_{c}(\mathbb{R})}D_{mS}(U;\mathcal{X})$

and $V\not\in\operatorname{Med}(\mathcal{X};D_{mS})$, which is a
contradiction.

The proof for $D_{FS}$ is analogous.

∎

##  6 Discussion

We characterized the set of $1$-medians as fuzzy sets such that their support
functions at $(u,\alpha)$ are univariate medians of the support function of
the fuzzy random number at $(u,\alpha)$ for any
$(u,\alpha)\in\mathbb{S}^{0}\times[0,1]$. Thanks to this characterization, the
set of $1$-medians coincides with what we define as the set of support
medians, which contains the main pre-existing notions of median for fuzzy
random numbers (for instance $\operatorname{med}_{Gr}$ in [17] and
$\operatorname{med}_{Si}$ in [35]). Since the $\rho_{1}$ metric is a natural
generalization to the fuzzy setting of the absolute value in $\mathbb{R}$, the
notion of $1$-median of a fuzzy random number in (9) is a natural
generalization to fuzzy numbers of the univariate median for real random
variables.

In [35] it is shown that, for every random fuzzy number, there exists at least
one fuzzy number which is a $1$-median in the sense of (9), denoted by
$\operatorname{med}_{Si}$. The main result of this manuscript is characterize
the set of fuzzy numbers which are $1$-medians in the sense of (9), the set of
support medians, collected in Theorem 4.2.

It is possible, in the univiariate case, that real random variables could have
infinitely many medians, while in the fuzzy setting there exist different
notions of unique median. The set of support medians contains, for every fuzzy
random number, the medians $\operatorname{med}_{Gr}$ and
$\operatorname{med}_{Si}$, indicating a good generalization of the concept of
median for fuzzy random numbers. This set also has, in some cases, infinitely
many other elements satisfying the minimization condition presented in (9).

The notion of depth arises from the idea of ordering the elements of a space
and generalizing the notion of median, as the element that maximizes the
depth. There exists some proposals of depth in the fuzzy setting, such as the
Tukey depth, $D_{FT}$, the fuzzy simplicial depths, $D_{mS}$ and $D_{FS}$, the
projection depth, $D_{FP}$, or the $L^{r}$-type depths, $D_{r}.$ The second
aim of this manuscript is to give a connection between the support medians of
a fuzzy random number and the different proposals of depth present in the
literature.

The main result of this manuscript states that the set of support medians and
the set of $1$-medians coincide. It implies that the set of elements which
maximize $D_{1}$ depth is the set of support medians (see Proposition 4.1).
Theorem 4.4 states that, for every fuzzy random number, the set of elements
which maximize the Tukey depth, $D_{FT}$, is the set of support medians and
therefore the set of $1$-medians. In the case of the projection depth, we may
assume that the fuzzy random number is non-degenerated. In that case, Theorem
4.8 assures that the unique element that maximizes $D_{FP}$ is
$\operatorname{med}_{Si}$. When we consider the simplicial depths, $D_{mS}$
and $D_{FS}$, we must consider fuzzy random numbers in
$C^{0}[\mathcal{F}_{c}(\mathbb{R}^{p})]$. Theorem 4.9 shows that the sets
which maximize $D_{mS}$ and $D_{FS}$ is, exactly, the set of support medians.
In the general case this results is not true, as it is shown in Example 4.10.

In this manuscript we give a solution for the minimization problem in (9) and
characterize the set of $1$-medians in terms of the depth proposals in the
fuzzy setting. On the one hand, this characterization shows that depth
functions in the fuzzy setting are good generalizations since one of the main
goals of depth is to generalize the concept of median. On the other hand,
$1$-medians of fuzzy random numbers could be computed via depth functions,
avoiding the minimization problem in (9).

For future work, it could be interesting to develop some techniques to compute
depth proposals, in order to deal with the set of $1$-medians, thanks to the
above results which connect the notion of median with the notion of depth for
the fuzzy setting.

As depth functions try to order the elements of a space with respect to a
distribution, it is also interesting to study the generalization, to the fuzzy
setting, of the notion of quantile via depth functions.

Acknowledgments The authors are supported by grant PID2022-139237NB-I00 funded
by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”.
Additionally, L. González was supported by the Spanish Ministerio de Ciencia,
Innovación y Universidades grant MTM2017-86061-C2-2-P. P. Terán is also
supported by the Ministerio de Ciencia, Innovación y Universidades grant
PID2019-104486GB-I00.

## References

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