#  New multivariate Gini’s indices

Marco Capaldo1 Jorge Navarro 2
1 Dipartimento di Matematica, Università degli Studi di Salerno
Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy
2 Departamento de Estadística e Investigación Operativa, Facultad de
Matemática
Universidad de Murcia, Murcia 30100, Spain
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID: 0000-0002-0255-8935ORCID:
0000-0003-2822-915X

(; December 14, 2023)

###### Abstract

The Gini’s mean difference was defined as the expected absolute difference
between a random variable and its independent copy. The corresponding
normalized version, namely Gini’s index, denotes two times the area between
the egalitarian line and the Lorenz curve. Both are dispersion indices because
they quantify how far a random variable and its independent copy are. Aiming
to measure dispersion in the multivariate case, we define and study new Gini’s
indices. For the bivariate case we provide several results and we point out
that they are “dependence-dispersion” indices. Covariance representations are
exhibited, with an interpretation also in terms of conditional distributions.
Further results, bounds and illustrative examples are discussed too.
Multivariate extensions are defined, aiming to apply both indices in more
general settings. Then, we define efficiency Gini’s indices for any semi-
coherent system and we discuss about their interpretation. Empirical versions
are considered in order as well to apply multivariate Gini’s indices to data.

Mathematics Subject Classification: 62H05, 91B82, 60E15.

Keywords: Gini’s index, Gini’s mean difference, Dispersion measure, Copula,
Bound, Coherent system.

##  1 Introduction

Information measures provide more specific details about random variables than
those given by mean values, variances, and others moments. In this sense, they
allow to broaden the knowledge about phenomena shaped by random variables. The
Gini’s index and the Gini’s mean difference are two popular dispersion
measures. The Gini’s index is a well known measure of income inequality in a
population related with the Lorenz curve. For its history and alternative
formulations we refer the reader to Arnold and Sarabia [4] and Ceriani and
Verme [8].

Given a nonnegative random variable $X$, with cumulative distribution function
(c.d.f.) $F(t)=\mathsf{Pr}(X\leq t)$ and survival function (s.f.)
$\bar{F}=1-F$, the Gini’s mean difference of $X$ was defined as

$\mathsf{GMD}(X)\coloneqq\mathsf{E}|X-X^{\prime}|=2\int_{0}^{+\infty}F(t)\bar{F}(t){\rm
d}t,$ (1.1)

where $X^{\prime}$ is an independent copy of $X$ (see, for instance, Arnold
and Sarabia [4] and Yitzhaki [36]). This information measure quantifies the
variability between two independent and identically distributed (i.i.d.)
random variables by using the absolute mean difference. In other words, the
Gini’s mean difference measures how far they are respect to the egalitarian
line, and in this sense it can be seen as dispersion index. A normalized
version of Eq. (1.1), namely Gini’s index of $X$, was defined as

$\mathsf{G}(X)\coloneqq\frac{\mathsf{GMD}(X)}{2\mathsf{E}(X)}\in[0,1]$ (1.2)

where $0<\mathsf{E}(X)<\infty$. It represents twice the area between the
egalitarian line and the Lorenz curve (see, for example, Gastwirth [15] and
Arnold and Sarabia [4], p. 52). We point out that the Gini’s indices given in
Eqs. (1.1) and (1.2) share many properties with the variance, since

$\mathsf{GMD}(X)=4\mathsf{Cov}(X,F(X)),\qquad\mathsf{G}(X)=\frac{2}{\mathsf{E}(X)}\mathsf{Cov}(X,F(X)).$
(1.3)

They can be informative about the properties of distributions that depart from
normality, see Yitzhaki [36]. For this and other reasons, Eq. (1.2) plays an
important role in economics and social sciences, when the normal distribution
does not provide a good approximation to data (see Yitzhaki [36] and
references therein).

Some generalizations and extensions of the Gini’s index have been considered
in the literature, see, for instance, Koshevoy and Mosler [17], Sections 5.3.2
and 7.4.1 in Arnold and Sarabia [4] and references therein. Moreover,
different versions of the Gini’s indices introduced above can be found in the
context of concordance measures. For example, if $(X,Y)$ has copula $C$, with
absolutely continuous marginal distributions, then the population version of
Gini’s measure of association for $X$ and $Y$ is given by

$\gamma_{C}=4\left(\int_{0}^{1}C(u,1-u){\rm d}u-\int_{0}^{1}(u-C(u,u)){\rm
d}u\right),$

see Section 5.1.4 of Nelsen [24] and Fuchs and Tschimpke [14]. Recently, the
copula-distorted Gini’s mean difference of $X$ has been defined in Capaldo et
al. [6], aiming to generalize Eq. (1.1) for the cases in which the random
variables involved are possibly dependent and where one s.f. (or c.d.f.) is
distorted from the other, making use of a distortion function in a parametric
family. Since the absolute mean difference can be view as a sort of distance
between the random variables involved, related minimizing problems of distance
can be found in Capaldo et al. [6] and Ortega-Jiménez et al. [25]. Moreover,
see Ortega-Jiménez et al. [26] for stochastic comparisons between absolute
mean differences.

In this paper for a random vector $(X,Y)$ we propose and study a bivariate
version of the Gini’s mean difference in Eq. (1.1), defined as
$\mathsf{GMD}(X,Y)=\mathsf{E}|X-Y|$. In this case the distribution of $Y$ is
not necessarily equal to the distribution of $X$ and they can be dependent. We
also define a bivariate version of the Gini’s index in Eq. (1.2). We extend
both Gini’s indices to the multivariate case. Moreover we focus on the meaning
of the new measures that represent “dependence-dispersion” indices. Along this
line, for fixed marginal distributions, we prove that the indices decrease
when the positive dependence increases. It is also related with the Schur-
concavity of the underline copula. Additionally, for fixed dependence we prove
that the indices decrease as the marginal dispersion decreases. We also show
how they can be informative about the reliability of a system and, in this
sense, we define their efficiency versions for any semi-coherent system.
Empirical versions are also considered aiming to provide applications for the
new Gini’s indices to data. We show some illustrative examples where we find
interpretations in terms of areas.

The paper is organized as follows. In Section 2 we recall useful notions
regarding stochastic orders, copula functions with related properties, and
some probabilistic inequalities. Section 3 is devoted to the bivariate Gini’s
mean difference and bivariate Gini’s index. After definitions we obtain
covariance representations for the new measures. Moreover, we show basic
properties and provide several results, by extending them to the multivariate
case. Section 4 contains bounds and inequalities for both bivariate Gini’s
indices. In Section 5 we define the efficiency Gini’s mean difference for any
semi-coherent system. We also define an efficiency version for the bivariate
Gini’s index and we discuss about its interpretation. Examples and simulations
are given in Section 6, by providing also empirical versions for the new
Gini’s indices.

##  2 Background and notations

In this section we fix the notation by recalling preliminary notions such as
usual stochastic and dispersive orders, copula functions and useful
probabilistic inequalities. Throughout the paper the terms increasing and
decreasing are used as “non-decreasing” and “non-increasing”, respectively. In
this paper we consider only nonnegative random variables, namely random
lifetime. However we must note that some results also holds without this
assumption.

For a random lifetime $X$ with c.d.f. $F$, the quantile function is denoted as
$F^{-1}(u)=\sup\\{x\colon F(x)\leq u\\}$, for $u\in[0,1]$. In addition, in the
absolutely continuous case, the probability density function (p.d.f.) is
defined as $f(x)=F^{\prime}(x)$ at points $x$ such that $F$ is derivable. We
assume that $0<\mu=\mathsf{E}(X)=\int_{0}^{+\infty}\bar{F}(t){\rm d}t<\infty$,
where $\bar{F}=1-F$.

We say that $X$ is smaller than $Y$ in the usual stochastic order, denoted by
$X\leq_{st}Y$, if and only if $\bar{F}_{X}(t)\leq\bar{F}_{Y}(t)$, for all
$t\geq 0$. If there is equality in law, then we write $X=_{st}Y$. We recall
that $X$ is said to be smaller than $Y$ in the dispersive order, denoted as
$X\leq_{d}Y$, if and only if

$F_{X}^{-1}(v)-F_{X}^{-1}(u)\geq
F_{Y}^{-1}(v)-F_{Y}^{-1}(u)\qquad\hbox{whenever }0<u\leq v<1.$

In addition, if $X$ and $Y$ are absolutely continuous, then

$X\leq_{d}Y\quad\hbox{if and only if}\quad f_{X}(F_{X}^{-1}(u))\geq
f_{Y}(F_{Y}^{-1}(u))\quad\forall\,u\in(0,1).$

While the usual stochastic order is used to compare two random variables in
size, the dispersive order compares their dispersion. Moreover, if
$X\leq_{st}Y$ then $\mathsf{E}(X)\leq\mathsf{E}(Y)$, while for $X\leq_{d}Y$
one has $\mathsf{Var}(X)\leq\mathsf{Var}(Y)$. For more details about
stochastic orders we refer the reader to Shaked and Shanthikumar [32].

From Sklar’s theorem a multivariate distribution can be written in terms of
the marginal distributions, making use of a suitable function, namely copula
function. Thus, given a random vector ${\bf X}=(X_{1},\dots,X_{n})$ with joint
c.d.f.

$F_{\bf X}(x_{1},\dots,x_{n})=\mathsf{Pr}(X_{1}\leq x_{1},\dots,X_{n}\leq
x_{n})$

and marginal c.d.f.’s $F_{X_{1}},\dots,F_{X_{n}}$, it follows

$F_{\bf X}(x_{1},\dots,x_{n})=C(F_{X_{1}}(x_{1}),\dots,F_{X_{n}}(x_{n})),$
(2.1)

for $x_{1},\dots,x_{n}\in\mathbb{R}$, where $C$ is called copula function of
${\bf X}$. If the marginal distributions are all continuous then the copula is
unique. Similarly, referring to the joint s.f.

$\bar{F}_{\bf
X}(x_{1},\dots,x_{n})=\mathsf{Pr}(X_{1}>x_{1},\dots,X_{n}>x_{n})$

and marginal s.f.’s $\bar{F}_{X_{1}},\dots,\bar{F}_{X_{n}}$, it follows

$\bar{F}_{\bf
X}(x_{1},\dots,x_{n})=\widehat{C}(\bar{F}_{X_{1}}(x_{1}),\dots,\bar{F}_{X_{n}}(x_{n})),$
(2.2)

for $x_{1},\dots,x_{n}\in\mathbb{R}$, where $\widehat{C}$ is called survival
copula function of ${\bf X}$. Moreover, the functions

$\delta(u)=C(u,\dots,u),\qquad\widehat{\delta}(u)=\widehat{C}(u,\dots,u),\qquad
u\in[0,1],$

represent the diagonal sections of $C$ and $\widehat{C}$, respectively.

We recall that ${\bf X}=(X_{1},\dots,X_{n})$ is said to be exchangeable (exc.)
if

$(X_{1},\dots,X_{n})=_{st}(X_{\sigma(1)},\dots,X_{\sigma(n)})$

for all permutations $\sigma$ of order $n$. From exchangeability it follows
that $X_{1},\dots,X_{n}$ are identically distributed (i.d.).

For simplicity in the bivariate case we now recall some properties regarding
copulas. First, for all $u,v\in[0,1]$ the relation between $C$ and
$\widehat{C}$ is given by

$\widehat{C}(u,v)=u+v-1+C(1-u,1-v).$ (2.3)

Moreover, for every copula $C$ and for all $u,v\in[0,1]$ one has

$W(u,v)\leq C(u,v)\leq M\\{u,v\\},$ (2.4)

where $W(u,v)=\max\\{u+v-1,0\\}$ and $M(u,v)=\min\\{u,v\\}$ are two copulas
called Fréchet-Hoeffding (FH) bounds (lower and upper, respectively). Another
important copula that we will encounter is the independence copula, usually
denoted as $\Pi(u,v)=uv$, for all $u,v\in[0,1]$. Moreover, we will also
consider the Farlie-Gumbel-Morgenstern (FGM) family of copulas, defined for
all $u,v\in[0,1]$ as

$C_{\theta}(u,v)=uv\left(1+\theta(1-u)(1-v)\right),\qquad\theta\in[-1,1].$
(2.5)

Clearly, from Eq. (2.5) one has $C_{0}(u,v)=\Pi(u,v)$. A family of copulas
that includes $M,\Pi$ and $W$ is called comprehensive. An example is the
Clayton family of copulas, defined for all $u,v\in[0,1]$ as

$C_{\theta}(u,v)=\left[\max(u^{-\theta}+v^{-\theta}-1,0)\right]^{-\frac{1}{\theta}},\qquad\theta\in[-1,0)\cup(0,+\infty),$
(2.6)

where $C_{-1}=W$, $C_{0}=\Pi$ and $C_{+\infty}=M$. Another example of
comprehensive family of copulas is represented by the Frank family of copulas,
defined for all $u,v\in[0,1]$ as

$C_{\theta}(u,v)=-\frac{1}{\theta}\ln\left(1-\frac{(1-e^{-\theta
u})(1-e^{-\theta
v})}{e^{-\theta}-1}\right),\qquad\theta\in(-\infty,0)\cup(0,+\infty),$ (2.7)

where $C_{-\infty}=W$, $C_{0}=\Pi$ and $C_{+\infty}=M$.

For our aims, we recall that a copula $C_{1}$ is smaller than another copula
$C_{2}$ under the concordance order, and write $C_{1}\prec C_{2}$, if
$C_{1}(u,v)\leq C_{2}(u,v)$ for all $u,v\in[0,1]$. Moreover, a random vector
$(X,Y)$ with a copula $C$ is said to be positively quadrant dependent (PQD),
if and only if $\Pi\prec C$. Conversely it has the negatively quadrant
dependent (NQD) property if and only if $C\prec\Pi$. Throughout the paper, we
will denote with $\partial_{1}{C}$ (respectively $\partial_{2}{C}$) the
partial derivative with respect to the first (second) argument of $C$. For
these and other basic properties of copulas we refer the reader to Durante and
Sempi [13] and Nelsen [24].

In addition, we now recall some properties for bivariate c.d.f.’s or s.f.’s,
that clearly are related with copulas and survival copulas from Eqs. (2.1) and
(2.2). Let $(X,Y)$ be a random vector with joint s.f. $\bar{H}$. We say that
$\bar{H}$ is Schur-concave (Schur-convex) if

$\bar{H}(x,y)\leq\bar{H}(\tilde{x},\tilde{y})\,(\geq),$ (2.8)

for all $x+y=\tilde{x}+\tilde{y}$ and
$\min(x,y)\leq\min(\tilde{x},\tilde{y})$, see Durante and Sempi [13], p. 278.
We remark that the Fréchet-Hoeffding lower bound in Eq. (2.4) represents the
only Schur-convex copula (see, e.g., Nelsen [24], p. 104). This properties can
be relaxed as follows: $\bar{H}$ is weakly Schur-concave (Schur-convex) if

$\bar{H}(x,y)\leq\bar{H}(z,z)\,(\geq),$ (2.9)

for all $x,y$ and $z=(x+y)/2$, see Durante and Papini [12]. Moreover,
$\bar{H}$ is Schur-constant if

$\bar{H}(x,y)=\bar{G}(x+y),$ (2.10)

for all $x,y\geq 0$, where $\bar{G}$ is a univariate s.f., see Caramellino and
Spizzichino [7]. We remark that in the case of Schur-constant $X$ and $Y$ are
exc. See also Pellerey and Navarro [27] for Schur-constant joint s.f. related
to monotonicity properties of dependent variables given their sum.

We conclude this section by recalling well known probabilistic inequalities,
useful for our aims. Given a random variable $Z$ and a convex function $\psi$,
the Jensen’s inequality states that

$\psi\left(\mathsf{E}(Z)\right)\leq\mathsf{E}\left(\psi(Z)\right).$ (2.11)

Moreover, if $Z$ is a random lifetime and $a>0$, the Markov’s inequality
states that

$\mathsf{Pr}\left(Z\geq a\right)\leq\frac{\mathsf{E}(Z)}{a}.$ (2.12)

##  3 Bivariate Gini’s indices

We now define a bivariate version of the Gini’s mean difference introduced in
Eq. (1.1), and we study its basic properties. The formal definition can be
stated as follows.

###### Definition 3.1

Let $(X,Y)$ be a random vector. The bivariate Gini’s mean difference of
$(X,Y)$ is defined as

$\mathsf{GMD}(X,Y)=\mathsf{E}|X-Y|.$ (3.1)

For our aims, we denote $U=\max\\{X,Y\\}$ and $L=\min\\{X,Y\\}$. By using
$|x-y|=\max\\{x,y\\}-\min\\{x,y\\}$ in Eq. (3.1), it follows

$\mathsf{GMD}(X,Y)=\mathsf{E}(U)-\mathsf{E}(L).$ (3.2)

Therefore, Eq. (3.2) can be seen as the expected distance between the
lifetimes of a parallel system and a series system, both having two components
distributed as $X$ and $Y$, respectively. Note that $X$ and $Y$ can be
dependent. Moreover, since $|x-y|=x+y-2\min\\{x,y\\}$, from Eq. (3.1) it also
follows

$\mathsf{GMD}(X,Y)=\mathsf{E}(X)+\mathsf{E}(Y)-2\mathsf{E}(L),$ (3.3)

and, by using Eqs. (3.2) and (3.3), one has the well known property
$\mathsf{E}(X)+\mathsf{E}(Y)=\mathsf{E}(U)+\mathsf{E}(L)$.

Suppose now that $(X,Y)$ has survival copula $\widehat{C}$, with marginal
s.f.’s $\bar{F}_{X},\bar{F}_{Y}$, respectively. Then, for all $t\geq 0$ one
has

$\bar{F}_{L}(t)=\mathsf{Pr}\left(X>t,Y>t\right)=\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))$

and therefore Eq. (3.3) can be rewritten as

$\mathsf{GMD}(X,Y)=\int_{0}^{+\infty}\left\\{\bar{F}_{X}(t)+\bar{F}_{Y}(t)-2\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\right\\}{\rm
d}t.$ (3.4)

Clearly, if $X$ and $Y$ are i.i.d. Eq. (3.4) is the univariate Gini’s mean
difference in Eq. (1.1). If $\bar{F}_{Y}(t)=h_{\alpha}(\bar{F}_{X}(t))$ for
all $t$, where $h_{\alpha}$ is a distortion function in a parametric family
with real value $\alpha$ and, by considering also a parametric family of
survival copulas for $(X,Y)$, Eq. (3.4) reduces to the copula-distorted Gini’s
mean difference defined in Capaldo et al. [6]. One can use the c.d.f.’s
configuration instead of the s.f.’s one, since from Eqs. (2.3) and (3.4) it
follows

$\mathsf{GMD}(X,Y)=\int_{0}^{+\infty}\left\\{F_{X}(t)+F_{Y}(t)-2C(F_{X}(t),F_{Y}(t))\right\\}{\rm
d}t,$ (3.5)

where $C$ denotes the copula function of $(X,Y)$, and $F_{X}$, $F_{Y}$ the
marginal c.d.f.’s.

As for the univariate case in Eq. (1.2), we now define the bivariate Gini’s
index as a ratio related to Eq. (3.1).

###### Definition 3.2

Let $(X,Y)$ be a random vector. The bivariate Gini’s index of $(X,Y)$ is
defined as

$\mathsf{G}(X,Y)=\frac{\mathsf{GMD}(X,Y)}{\mathsf{E}(X)+\mathsf{E}(Y)}.$ (3.6)

Clearly, if $X$ and $Y$ are i.i.d., Eq. (3.6) is the univariate Gini’s index
in Eq. (1.2). We remark that Eq. (3.6) extends the copula-distorted Gini’s
index defined in [6]. Recalling Eq. (3.2), one has

$\mathsf{G}(X,Y)=\frac{\mathsf{E}(U)-\mathsf{E}(L)}{\mathsf{E}(U)+\mathsf{E}(L)}.$
(3.7)

Hence $\mathsf{G}(X,Y)\in[0,1]$, with $\mathsf{G}(X,Y)=0$ if and only if
$\mathsf{E}(U)=\mathsf{E}(L)$. In addition, $\mathsf{G}(X,Y)=1$ when
$\mathsf{E}(L)=0$.

As for the univariate case (see Eq. (1.3)), we now explore the possibility to
express the bivariate Gini’s indices in terms of a covariance between the
random lifetimes involved and their suitable transformations based on
distributions. For this reason, given a random vector $(X,Y)$ we define the
following functions

$\gamma_{1}(t)\coloneqq{\mathsf{Pr}\left(Y>t|X=t\right)},\qquad\gamma_{2}(t)\coloneqq{\mathsf{Pr}\left(X>t|Y=t\right)},\quad
t\geq 0.$ (3.8)

If $\widehat{C}$ denotes the survival copula of $(X,Y)$, then it is easy to
get (see, for instance, Navarro and Sordo [21] or Nelsen [24], p. 217)

$\gamma_{1}(t)=\partial_{1}\widehat{C}\left(\bar{F}_{X}(t),\bar{F}_{Y}(t)\right),\qquad\gamma_{2}(t)=\partial_{2}\widehat{C}\left(\bar{F}_{X}(t),\bar{F}_{Y}(t)\right),\quad
t\geq 0.$ (3.9)

In the next result we first obtain a covariance representation of
$\mathsf{E}(L)$.

###### Proposition 3.1

Let $(X,Y)$ be a random vector with absolutely continuous distribution. Then

$\mathsf{E}(L)=\mathsf{Cov}\left(X,\gamma_{1}(X)\right)+\mathsf{Cov}\left(Y,\gamma_{2}(Y)\right)+\mathsf{E}(X)\mathsf{E}(\gamma_{1}(X))+\mathsf{E}(Y)\mathsf{E}(\gamma_{2}(Y)).$
(3.10)

  * Proof.

Since $\bar{F}_{L}(t)=\widehat{C}\left(\bar{F}_{X}(t),\bar{F}_{Y}(t)\right)$,
the p.d.f. of $L$ is

$f_{L}(t)=f_{X}(t)\partial_{1}\widehat{C}\left(\bar{F}_{X}(t),\bar{F}_{Y}(t)\right)+f_{Y}(t)\partial_{2}\widehat{C}\left(\bar{F}_{X}(t),\bar{F}_{Y}(t)\right),\qquad
t\geq 0,$

where $f_{X}$ and $f_{Y}$ are the marginal p.d.f.’s. Therefore, recalling Eq.
(3.9), one has

$\mathsf{E}(L)=\mathsf{E}(X\gamma_{1}(X))+\mathsf{E}(Y\gamma_{2}(Y)).$

Finally, Eq. (3.10) follows from the definition of covariance. $\Box$

If $X$ and $Y$ are absolutely continuous, then from Eq. (3.8) one has
$\mathsf{E}(\gamma_{1}(X))=\mathsf{Pr}\left(Y>X\right)$ and
$\mathsf{E}(\gamma_{2}(Y))=1-\mathsf{E}(\gamma_{1}(X))$. Therefore Eq. (3.10)
can also be written as

$\mathsf{E}(L)=\mathsf{Cov}\left(X,\gamma_{1}(X)\right)+\mathsf{Cov}\left(Y,\gamma_{2}(Y)\right)+\left(\mathsf{E}(X)-\mathsf{E}(Y)\right)\mathsf{Pr}\left(Y>X\right)+\mathsf{E}(Y).$
(3.11)

Along the same line, in the next result we obtain a covariance representation
for the bivariate Gini’s mean difference. Recalling Eq. (3.6), dividing by
$\mathsf{E}(X)+\mathsf{E}(Y)$ one can obtain a covariance representation for
the bivariate Gini’s index.

###### Theorem 3.1

Let $(X,Y)$ be a random vector with absolutely continuous distribution. Then

$\mathsf{GMD}(X,Y)=2\left[\mathsf{E}(X)-\mathsf{E}(Y)\right]\left[\frac{1}{2}-\mathsf{Pr}(Y>X)\right]-2\mathsf{Cov}(X,\gamma_{1}(X))-2\mathsf{Cov}(Y,\gamma_{2}(Y)).$

  * Proof.

The thesis immediately follows from Eqs. (3.3) and (3.11). $\Box$

###### Corollary 3.1

Under the same assumptions of Theorem 3.1:

(i)

if $\mathsf{E}(X)=\mathsf{E}(Y)$ or if $\mathsf{Pr}(Y>X)=1/2$, then

$\mathsf{GMD}(X,Y)=-2\mathsf{Cov}(X,\gamma_{1}(X))-2\mathsf{Cov}(Y,\gamma_{2}(Y));$

(ii)

if $X$ and $Y$ are exc., then

$\mathsf{GMD}(X,Y)=-4\mathsf{Cov}(X,\gamma_{1}(X)).$ (3.12)

From the left-hand-side of Eq. (3.8), we denote with
$F_{2|1}(t|t)=\mathsf{Pr}\left(Y\leq t|X=t\right)=1-\gamma_{1}(t)$, for all
$t\geq 0$, the probability that the second lifetime is less or equal to $t$
conditioned by the fact that the first lifetime is equal to $t$. If $X$ and
$Y$ are exc., from Eqs. (3.6) and (3.12), it follows

$\mathsf{G}(X,Y)=\frac{2}{\mathsf{E}(X)}\mathsf{Cov}(X,F_{2|1}(X|X))=-\frac{2}{\mathsf{E}(X)}\mathsf{Cov}(X,\gamma_{1}(X)),$
(3.13)

that extends the covariance representation of the univariate Gini’s index in
Eq. (1.3). Under the exchangeability assumption, Eq. (3.13) represents another
way to compute the bivariate Gini’s index making use of conditional
distributions instead of copula representations.

As an example, given a random vector $(X,Y)$, let us consider the exponential
conditional distributions with joint p.d.f. given by

$f(x,y)=\exp\\{-(c_{11}+c_{12}x+c_{21}y+c_{22}xy)\\},\qquad x>0,y>0,$

where $c_{12}>0$, $c_{21}>0$, $c_{22}\geq 0$ (the case $c_{22}=0$ corresponds
to the independent case) and

$c_{11}=\ln\left[\frac{1}{c_{12}c_{21}}\int_{0}^{+\infty}e^{-u}\left(1+\frac{c_{22}u}{c_{12}c_{21}}\right)^{-1}{\rm
d}u\right].$ (3.14)

Therefore one has

$\mathsf{Pr}\left(Y>y|X=x\right)=\exp\\{-(c_{21}+c_{22}x)y\\},\qquad x>0,y>0,$

(see Arnold et al. [2] and [3]). For instance, if $c_{12}=c_{21}=c_{22}=1$,
from Eq. (3.14) one has $c_{11}=-0.516932$. Hence, recalling Eq. (3.8), it
follows

$\gamma_{1}(t)=\exp\\{-(t+t^{2})\\},\qquad t>0.$

Under these assumptions, $X$ and $Y$ are exc. and one numerically has
$\mathsf{E}(X)=0.676875$, $\mathsf{E}(\gamma_{1}(X))=0.5$ and
$\mathsf{E}(X\gamma_{1}(X))=0.135428$. Therefore, from the definition of
covariance and Eq. (3.13), one obtains $\mathsf{G}(X,Y)=0.599843$.

In the next result we show some basic properties satisfied by the bivariate
Gini’s mean difference.

###### Theorem 3.2

The bivariate Gini’s mean difference of $(X,Y)$ satisfies the following
properties:

(i) (law invariance) if $(X,Y)=_{st}(\tilde{X},\tilde{Y})$, then
$\mathsf{GMD}(X,Y)=\mathsf{GMD}(\tilde{X},\tilde{Y})$;

(ii) (translation invariance)
$\mathsf{GMD}(X+\lambda,Y+\lambda)=\mathsf{GMD}(X,Y)$, for all
$\lambda\in\mathbb{R}$;

(iii) (homogeneity) $\mathsf{GMD}(\lambda X,\lambda
Y)=|\lambda|\mathsf{GMD}(X,Y)$, for all $\lambda\in\mathbb{R}$;

(iv) (non-negativity) $\mathsf{GMD}(X,Y)\geq 0$ for all $X$ and $Y$. In
addition if $X$ and $Y$ are i.d. and $C(u,v)=\min\\{u,v\\}$, for all
$u,v\in[0,1]$, then $\mathsf{GMD}(X,Y)=0$. Conversely, if
$\mathsf{GMD}(X,Y)=0$, then $F_{X}=F_{Y}$ and if $F_{X}$ is continuous then
$C(u,v)=\min\\{u,v\\}$, for all $u,v\in[0,1]$.

  * Proof.

The property (i) is trivial. The properties (ii) and (iii) immediately follows
from Eq. (3.1). $\mathsf{GMD}(X,Y)\geq 0$ also holds trivially from Eq. (3.1).
If $X$ and $Y$ are i.d. and $C(u,v)=\min\\{u,v\\}$, for all $u,v\in[0,1]$,
then $\mathsf{GMD}(X,Y)=0$ from Eq. (3.5). Conversely, if
$\mathsf{GMD}(X,Y)=0$, from Eq. (3.2) one has $\mathsf{E}(U)=\mathsf{E}(L)$
and since $L\leq_{st}U$, from Theorem 1.A.8. in [32], it follows
$\bar{F}_{L}=\bar{F}_{U}$. Therefore, since $L\leq_{st}X\leq_{st}U$ and
$L\leq_{st}Y\leq_{st}U$ then $F_{X}=F_{Y}$. Under these assumptions and if
$F_{X}$ is continuous, from Eq. (3.5) one has $\delta(u)=u$ for all
$u\in[0,1]$. Therefore for all $u,v\in[0,1]$

$C(u,v)\geq
C(\min\\{u,v\\},\min\\{u,v\\})=\delta(\min\\{u,v\\})=\min\\{u,v\\},$

and, by recalling $M$ in Eq. (2.4), finally the thesis holds. $\Box$

Similarly, from Eq. (3.6), it is easy to see that the bivariate Gini’s index
satisfies the following properties:

(i) (law invariance) if $(X,Y)=_{st}(\tilde{X},\tilde{Y})$, then
$\mathsf{G}(X,Y)=\mathsf{G}(\tilde{X},\tilde{Y})$;

(ii) (translation)
$\mathsf{G}(X+\lambda,Y+\lambda)=\frac{\mathsf{E}(X)+\mathsf{E}(Y)}{\mathsf{E}(X)+\mathsf{E}(Y)+2\lambda}\mathsf{G}(X,Y)$,
for all $\lambda\in\mathbb{R}$;

(iii) (homogeneity invariance) $\mathsf{G}(\lambda X,\lambda
Y)=\mathsf{G}(X,Y)$, for all $\lambda\in\mathbb{R}^{+}$;

(iv) (non-negativity) $\mathsf{G}(X,Y)\geq 0$ for all $X$ and $Y$. In addition
$\mathsf{G}(X,Y)=0$ under the same conditions given in (iv) of Theorem 3.2 for
$\mathsf{GMD}(X,Y)$.

Aiming to compare different bivariate Gini’s mean difference, below we provide
some results under suitable assumptions on the marginal distributions and
copula functions involved in Eqs. (3.4) and (3.5). First we show a dependence
ordering for the bivariate Gini’s mean difference and bivariate Gini’s index
through concordance order between two copula functions when we maintain the
marginals.

###### Proposition 3.2

Let $(X,Y)$ and $(\tilde{X},\tilde{Y})$ be two random vectors with copulas
$C_{1}$ and $C_{2}$, respectively. Suppose that $X=_{st}\tilde{X}$ and
$Y=_{st}\tilde{Y}$, with c.d.f.’s $F_{X}$ and $F_{Y}$, respectively. If
$C_{1}\prec C_{2}$, then

$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(\tilde{X},\tilde{Y}),\qquad\mathsf{G}(X,Y)\geq\mathsf{G}(\tilde{X},\tilde{Y}).$

  * Proof.

Under these assumptions, recalling Eq. (3.5), one has

$\mathsf{GMD}(X,Y)-\mathsf{GMD}(\tilde{X},\tilde{Y})=2\int_{0}^{+\infty}\left\\{C_{2}\left(F_{X}(t),F_{Y}(t)\right)-C_{1}\left(F_{X}(t),F_{Y}(t)\right)\right\\}{\rm
d}t.$ (3.15)

Hence, recalling also Eq. (3.6), the thesis follows from $C_{1}\prec C_{2}$.
$\Box$

Therefore if we fix the marginals and the dependence increases in the
concordance order, then $\mathsf{GMD}(X,Y)$ and $\mathsf{G}(X,Y)$ decrease,
that is, these dispersions indices decrease when the positive dependence
increases. When all the marginals are i.d. we have the following result.

###### Proposition 3.3

Let $(X,Y)$ and $(\tilde{X},\tilde{Y})$ be two random vectors with diagonal
sections $\delta_{1}$ and $\delta_{2}$, respectively. Suppose that all the
marginals are i.d. with common c.d.f. $F$. If $\delta_{1}\leq\delta_{2}$, then

$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(\tilde{X},\tilde{Y}),\qquad\mathsf{G}(X,Y)\geq\mathsf{G}(\tilde{X},\tilde{Y}),$

with equality reached when $\delta_{1}=\delta_{2}$.

  * Proof.

The thesis immediately follows from Eq. (3.15) when $X=_{st}Y$. $\Box$

Making use of Corollary 8 in Ortega-Jiménez et al. [26] and Proposition 3.3,
we can go further and get the following result.

###### Theorem 3.3

Let $(X,\tilde{X})$ and $(Y,\tilde{Y})$ be two random vectors with diagonal
sections $\delta_{1}$ and $\delta_{2}$, respectively. Suppose that
$X=_{st}\tilde{X}$ and $Y=_{st}\tilde{Y}$. If $Y\leq_{d}X$ and
$\delta_{1}\leq\delta_{2}$, then

$\mathsf{GMD}(X,\tilde{X})\geq\mathsf{GMD}(Y,\tilde{Y}).$

  * Proof.

Consider $(Y_{1},\tilde{Y}_{1})$ with diagonal section $\delta_{1}$ and
suppose that $Y=_{st}Y_{1}=_{st}\tilde{Y}_{1}$. The thesis follows since from
Corollary 8 in Ortega-Jiménez et al. [26] and Proposition 3.3 one has,
respectively,

$\mathsf{GMD}(X,\tilde{X})\geq\mathsf{GMD}(Y_{1},\tilde{Y}_{1})\geq\mathsf{GMD}(Y,\tilde{Y}).$

$\Box$

In the following result we compare the bivariate Gini’s mean difference in the
cases of homogeneous and heterogeneous marginal distributions.

###### Proposition 3.4

Let $(X,Y)$ be a random vector with survival copula $\widehat{C}$ and strictly
increasing diagonal section $\widehat{\delta}$. Let $(Z,\tilde{Z})$ be another
random vector with the same survival copula and common marginal s.f.
$\bar{G}$. If

$\widehat{\delta}^{-1}(\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t)))\leq\bar{G}(t)\leq\frac{\bar{F}_{X}(t)+\bar{F}_{Y}(t)}{2}$
(3.16)

holds for all $t$, then $\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(Z,\tilde{Z})$.

  * Proof.

From Eqs. (3.4) and (3.16), as
$\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t)))\leq\widehat{\delta}(\bar{G}(t))$,
one has

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle=2\int_{0}^{\infty}\left(\frac{\bar{F}_{X}(t)+\bar{F}_{Y}(t)}{2}-\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\right){\rm
d}t$ $\displaystyle\geq
2\int_{0}^{\infty}\left(\bar{G}(t)-\widehat{\delta}(\bar{G}(t))\right){\rm
d}t$ $\displaystyle=\mathsf{GMD}(Z,\tilde{Z})$

and the proof is complete. $\Box$

Note that both
$\bar{G}_{1}(t):=\widehat{\delta}^{-1}(\widehat{C}(\bar{F}_{X}(t),\bar{F}_{X}(t)))$
and $\bar{G}_{2}(t):=(\bar{F}_{X}(t)+\bar{F}_{Y}(t))/2$ are s.f.’s obtained as
distortions of $\bar{F}_{X}$ and $\bar{F}_{Y}$. We also remark that the
bivariate Gini’s mean difference decreases when the dependence does not change
and the marginal distributions are homogeneous, that is $Z$ and $\tilde{Z}$
are more similar than the original $X$ and $Y$. For example, when
$\widehat{C}=\Pi$, then we can apply Proposition 3.4 if $\bar{G}$ is between
the geometric and the arithmetic means of $\bar{F}_{X}$ and $\bar{F}_{Y}$,
since the condition expressed by Eq. (3.16) becomes

$\bar{G}_{1}(t)=\sqrt{\bar{F}_{X}(t)\bar{F}_{Y}(t)}\leq\bar{G}(t)\leq\bar{G}_{2}(t)=\frac{\bar{F}_{X}(t)+\bar{F}_{Y}(t)}{2}.$

Note that we can also apply Proposition 3.4 to both $\bar{G}_{1}$ and
$\bar{G}_{2}$. A similar result can be stated for the c.d.f. configuration
from Eq. (3.5), as follows.

###### Proposition 3.5

Let $(X,Y)$ be a random vector with copula $C$ and strictly increasing
diagonal section $\delta$. Let $(Z,\tilde{Z})$ be another random vector with
the same copula and common marginal c.d.f. $G$. If

$\delta^{-1}(C(F_{X}(t),F_{Y}(t)))\leq G(t)\leq\frac{F_{X}(t)+F_{Y}(t)}{2}$

holds for all $t$, then $\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(Z,\tilde{Z})$.

We can also obtain a result in order to compare heterogeneous cases. It can be
stated as follows. A similar result holds for the c.d.f. configuration given
in Eq. (3.5).

###### Proposition 3.6

Let $(X,Y)$ be a random vector with survival copula $\widehat{C}$. Let
$(\tilde{X},\tilde{Y})$ be another random vector with the same survival
copula. If
$\mathsf{E}(X)+\mathsf{E}(Y)\geq\mathsf{E}(\tilde{X})+\mathsf{E}(\tilde{Y})$
and

$\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\leq\widehat{C}(\bar{F}_{\tilde{X}}(t),\bar{F}_{\tilde{Y}}(t))$

holds for all $t$, then
$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(\tilde{X},\tilde{Y})$.

We remark that, under the assumptions of Proposition 3.6, when
$\mathsf{E}(X)+\mathsf{E}(Y)=\mathsf{E}(\tilde{X})+\mathsf{E}(\tilde{Y})$ one
also has $\mathsf{G}(X,Y)\geq\mathsf{G}(\tilde{X},\tilde{Y})$. Moreover,
recalling Eq. (2.8), if the survival copula is Schur-concave we can go further
and get the following result.

###### Proposition 3.7

Let $(X,Y)$ be a random vector with survival copula $\widehat{C}$. Let
$(\tilde{X},\tilde{Y})$ be another random vector with the same survival
copula. If $\widehat{C}$ is Schur-concave,
$\bar{F}_{X}+\bar{F}_{Y}=\bar{F}_{\tilde{X}}+\bar{F}_{\tilde{Y}}$ and
$\min(\bar{F}_{X},\bar{F}_{Y})\leq\min(\bar{F}_{\tilde{X}},\bar{F}_{\tilde{Y}})$,
then

$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(\tilde{X},\tilde{Y}),\qquad\mathsf{G}(X,Y)\geq\mathsf{G}(\tilde{X},\tilde{Y}).$

  * Proof.

Under the stated assumptions, if $\widehat{C}$ is Schur-concave, then

$\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\leq\widehat{C}(\bar{F}_{\tilde{X}}(t),\bar{F}_{\tilde{Y}}(t)).$

Therefore

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle=2\int_{0}^{\infty}\left(\frac{\bar{F}_{X}(t)+\bar{F}_{Y}(t)}{2}-\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\right){\rm
d}t$ $\displaystyle\geq
2\int_{0}^{\infty}\left(\frac{\bar{F}_{\tilde{X}}(t)+\bar{F}_{\tilde{Y}}(t)}{2}-\widehat{C}(\bar{F}_{\tilde{X}}(t),\bar{F}_{\tilde{Y}}(t))\right){\rm
d}t$ $\displaystyle=\mathsf{GMD}(\tilde{X},\tilde{Y})$

and the proof is complete. The result for the bivariate Gini’s index holds
since
$\mathsf{E}(X)+\mathsf{E}(Y)=\mathsf{E}(\tilde{X})+\mathsf{E}(\tilde{Y})$.
$\Box$

We remark that, for $\widehat{C}$ Schur-concave, if
$\bar{F}_{X}+\bar{F}_{Y}=\bar{F}_{\tilde{X}}+\bar{F}_{\tilde{Y}}$ and
$X\leq_{st}\tilde{X}\leq_{st}\tilde{Y}\leq_{st}Y$ hold, then

$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(\tilde{X},\tilde{Y}),\qquad\mathsf{G}(X,Y)\geq\mathsf{G}(\tilde{X},\tilde{Y}).$

Thus, for a fixed copula, if one increases the dispersions between the random
lifetimes involved in the vector, then the bivariate Gini’s indices increase.

The result included in Proposition 3.7 is related with Parrondo’s paradox in
reliability stated in Di Crescenzo [9] and in Navarro and Spizzichino [22]. In
this sense we know that many copulas are Schur-concave (for example all the
Archimedean copulas). Hence the preceding result holds for many dependence
models. If we just compare with the homogeneous case, we can relax the
condition of the copula to the weakly Schur-concavity defined in Eq. (2.9).

###### Proposition 3.8

Let $(X,Y)$ be a random vector with survival copula $\widehat{C}$. Let
$(Z,\tilde{Z})$ be another random vector with the same survival copula and
common marginal s.f. $\bar{G}$. If $\widehat{C}$ is weakly Schur-concave
(respectively Schur-convex) and $\bar{G}=(\bar{F}_{X}+\bar{F}_{Y})/2$, then

$\mathsf{GMD}(X,Y)\geq\mathsf{GMD}(Z,\tilde{Z})\,(\leq),\qquad\mathsf{G}(X,Y)\geq\mathsf{G}(Z,\tilde{Z})\,(\leq).$

  * Proof.

Under the stated assumptions, if $\widehat{C}$ is weakly Schur-concave, then

$\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\leq\widehat{C}(\bar{G}(t),\bar{G}(t)).$

Therefore

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle=2\int_{0}^{\infty}\left(\frac{\bar{F}_{X}(t)+\bar{F}_{Y}(t)}{2}-\widehat{C}(\bar{F}_{X}(t),\bar{F}_{Y}(t))\right){\rm
d}t$ $\displaystyle\geq
2\int_{0}^{\infty}\left(\bar{G}(t)-\widehat{C}(\bar{G}(t),\bar{G}(t))\right){\rm
d}t$ $\displaystyle=\mathsf{GMD}(Z,\tilde{Z})$

and the thesis follows. The proof for weakly Schur-convex copulas is similar.
The result for the bivariate Gini’s index holds since
$\mathsf{E}(X)+\mathsf{E}(Y)=2\mathsf{E}(Z)$. $\Box$

Recalling Eq. (2.10), in the following we consider the Schur-constant
assumption for $\widehat{C}$.

###### Proposition 3.9

Let $\widehat{C}$ be the survival copula of $(X,Y)$. If $\widehat{C}$ is
Schur-constant, then

$\mathsf{GMD}(X,Y)=\mathsf{E}(X),\qquad\mathsf{G}(X,Y)=\frac{1}{2}.$ (3.17)

  * Proof.

If $\widehat{C}$ is Schur-constant, then $|X-Y|=_{st}X$, as shown in Theorem 4
of Nelsen [23]. Therefore, the thesis follows recalling Eqs. (3.1) and (3.6).
$\Box$

Clearly, from the left-hand-side of Eq. (3.17), for $(X,Y)\sim\widehat{C}_{1}$
and $(\tilde{X},\tilde{Y})\sim\widehat{C}_{2}$ with Schur-constant copulas, if
$X\leq_{st}\tilde{X}$, then
$\mathsf{GMD}(X,Y)\leq\mathsf{GMD}(\tilde{X},\tilde{Y})$.

We conclude this section by extending the bivariate Gini’s indices to the
multivariate case. Given an $n$-dimensional random vector
$(X_{1},\dots,X_{n})$, we use the following notation

$X_{1:n}=\min\\{X_{1},\dots,X_{n}\\}\qquad\text{and}\qquad
X_{n:n}=\max\\{X_{1},\dots,X_{n}\\}.$

###### Definition 3.3

Let ${\bf X}=(X_{1},\dots,X_{n})$ be an $n$-dimensional random vector. The
multivariate Gini’s mean difference of ${\bf X}$ is defined as

$\mathsf{GMD}({\bf X})=\mathsf{GMD}(X_{1:n},X_{n:n}).$ (3.18)

Moreover, by denoting with $R\coloneqq X_{n:n}-X_{1:n}\geq 0$, the range of
the random vector ${\bf X}$, one has $\mathsf{GMD}({\bf
X})=\mathsf{E}(X_{n:n})-\mathsf{E}(X_{1:n})=\mathsf{E}(R)$. Clearly
$\mathsf{GMD}({\bf X})=0$ if and only if $X_{1}\equiv\dots\equiv X_{n}$. In
this sense the multivariate Gini’s mean difference can be used to measure the
dispersion of the random vector ${\bf X}$.

According to Eq. (3.6), one can define a multivariate ratio based on Eq.
(3.18), namely multivariate Gini’s index of ${\bf X}$, as follows

$\mathsf{G}({\bf X})\coloneqq\frac{\mathsf{GMD}({\bf
X})}{\mathsf{E}(X_{1:n})+\mathsf{E}(X_{n:n})}\in[0,1].$

If $X_{1},\dots,X_{n}$ are i.d., with common s.f. $\bar{F}$, having copula $C$
and survival copula $\widehat{C}$, Eq. (3.18) becomes

$\mathsf{GMD}({\bf
X})=\int_{0}^{+\infty}\left\\{1-\delta\left(F(t)\right)-{\hat{\delta}}\left(\bar{F}(t)\right)\right\\}{\rm
d}t,$ (3.19)

where $\delta$ and $\hat{\delta}$ are the diagonal sections of $C$ and
$\widehat{C}$, respectively. Moreover, if $X_{1},\dots,X_{n}$ are i.i.d.,
having common s.f. $\bar{F}$, then Eq. (3.19) reduces to

$\mathsf{GMD}({\bf
X})=\int_{0}^{+\infty}\left\\{1-\left(F(t)\right)^{n}-\left(\bar{F}(t)\right)^{n}\right\\}{\rm
d}t.$ (3.20)

###### Remark 3.1

If $X_{i}\sim\mathcal{U}(0,1)$, for $i\in\\{1,\dots,n\\}$, and they are
independent, then Eq. (3.20) becomes

$\mathsf{GMD}({\bf X})=\int_{0}^{1}\left\\{1-t^{n}-(1-t)^{n}\right\\}{\rm
d}t=\frac{n-1}{n+1}.$

Hence, $\mathsf{GMD}({\bf X})\rightarrow 1$ as $n\rightarrow+\infty$. For
instance, when $X_{1},\dots,X_{n}$ describe the lifetimes of $n$ components in
a system, then $\mathsf{GMD}({\bf X})$ goes to $1$ when the number of the
components increases significantly. In addition, since in this case
$\mathsf{E}(X_{n:n})=n/(n+1)$ and $\mathsf{E}(X_{1:n})=1/(n+1)$, it follows
that $\mathsf{GMD}({\bf X})=\mathsf{G}({\bf X})$. As examples, for $n=2,3,4,5$
one has $\mathsf{G}({\bf X})=\frac{1}{3},\frac{1}{2},\frac{3}{5},\frac{2}{3}$,
respectively.

###### Remark 3.2

If $X_{i}\sim{\rm Exp}(1)$, for $i\in\\{1,\dots,n\\}$, and they are
independent, by setting $t=e^{-x}$, Eq. (3.20) becomes

$\mathsf{GMD}({\bf
X})=\int_{0}^{1}\frac{\left\\{1-(1-t)^{n}-t^{n}\right\\}}{t}{\rm
d}t=H(n)-\frac{1}{n},$

where $H(n)=\sum_{k=1}^{n}\frac{1}{k}$ is the truncation of the harmonic
series. One has $\mathsf{GMD}({\bf X})\rightarrow+\infty$ as
$n\rightarrow+\infty$. In addition, it follows

$\mathsf{G}({\bf X})=\frac{nH(n)-1}{nH(n)+1},$

that goes to $1$ as $n$ increases. As examples, for $n=2,3,4,5$ one has
$\mathsf{G}({\bf X})=\frac{1}{2},\frac{9}{13},\frac{11}{14},\frac{125}{149}$,
respectively.

##  4 Bounds and inequalities

This section is devoted to show bounds and inequalities for the bivariate
Gini’s indices defined in Section 3. Bounds for the univariate Gini’s indices
given in Eqs. (1.1) and (1.2) can be found, for instance, in Yin et al. [35].
In the next result we use the Jensen’s inequality in Eq. (2.11) aiming to
provide lower bounds for Eqs. (3.1) and (3.6).

###### Proposition 4.1

Let $(X,Y)$ be a random vector. Then

$\mathsf{GMD}(X,Y)\geq|\mathsf{E}(X)-\mathsf{E}(Y)|,\qquad\mathsf{G}(X,Y)\geq\frac{|\mathsf{E}(X)-\mathsf{E}(Y)|}{\mathsf{E}(X)+\mathsf{E}(Y)}.$
(4.1)

  * Proof.

The thesis immediately follows by recalling Eqs. (3.1) and (3.6) and making
use of Eq. (2.11) with $\psi$ equal to the absolute value function and
$Z=X-Y$. $\Box$

Clearly, if $\mathsf{E}(X)\leq\mathsf{E}(Y)$, then from Eq. (4.1)

$\mathsf{GMD}(X,Y)\geq\mathsf{E}(Y)-\mathsf{E}(X),\qquad\mathsf{G}(X,Y)\geq\frac{\mathsf{E}(Y)-\mathsf{E}(X)}{\mathsf{E}(Y)+\mathsf{E}(X)}.$
(4.2)

###### Remark 4.1

Let $X$ and $Y$ be exponentially distributed having s.f.’s
$\bar{F}_{X}(t)=e^{-\lambda_{1}t}$ and $\bar{F}_{Y}(t)=e^{-\lambda_{2}t}$, for
$t\geq 0$, respectively. If $\lambda_{1}>\lambda_{2}>0$, from Eq. (4.2) one
has

$\mathsf{GMD}(X,Y)\geq\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}\lambda_{2}},\qquad\mathsf{G}(X,Y)\geq\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}}.$

Making use of the Markov’s inequality in Eq. (2.12), below we interpret
$\mathsf{GMD}(X,Y)$ as a bound for the probability that $Y$ assumes values in
a neighborhood of $X$.

###### Proposition 4.2

Let $(X,Y)$ be a random vector. For $a>0$, one has

$\mathsf{Pr}\left(X-a<Y<X+a\right)\geq 1-\frac{\mathsf{GMD}(X,Y)}{a}.$ (4.3)

  * Proof.

The thesis follows making use of Eq. (2.12) with $Z=|X-Y|$, and by recalling
Eq. (3.1). $\Box$

In addition, one can use the Markov’s inequality in Eq. (2.12) in order to
compare the bivariate Gini’s mean difference, for i.d. marginals, with the
univariate one, as shown in the next result.

###### Theorem 4.1

Let $(X,Y)$ be a random vector having i.d. marginals with mean $\mu$. One has

$\mathsf{Pr}\left(\frac{X}{\mu}-2\mathsf{G}(X)<\frac{Y}{\mu}<\frac{X}{\mu}+2\mathsf{G}(X)\right)\geq
1-\frac{\mathsf{G}(X,Y)}{\mathsf{G}(X)}.$

  * Proof.

Since $X=_{st}Y$, recalling Eqs. (1.1), (1.2), (3.1) and (3.6), one has

$\frac{\mathsf{GMD}(X,Y)}{\mathsf{GMD}(X)}=\frac{\mathsf{G}(X,Y)}{\mathsf{G}(X)}.$

Therefore, the thesis follows from Eq. (4.3) for $a=\mathsf{GMD}(X)$ and
dividing by $\mu$ in the inequalities stated in the left-hand-side. $\Box$

In the next result, making use of the FH bounds in Eq. (2.4) and Proposition
3.2, we obtain attainable bounds for the bivariate Gini’s mean difference. The
bounds for the bivariate Gini’s index can be obtained dividing by
$\mathsf{E}(X)+\mathsf{E}(Y)$.

###### Proposition 4.3

Let $(X,Y)$ be a random vector with marginal s.f.’s $\bar{F}_{X}$ and
$\bar{F}_{Y}$. Then

$\int_{0}^{+\infty}|\bar{F}_{X}(t)-\bar{F}_{Y}(t)|{\rm
d}t\leq\mathsf{GMD}(X,Y)\leq\int_{0}^{+\infty}\left\\{1-|\bar{F}_{X}(t)+\bar{F}_{Y}(t)-1|\right\\}{\rm
d}t,$ (4.4)

with bounds reached when the copula of $(X,Y)$ coincides with the FH bounds in
Eq. (2.4).

  * Proof.

By recalling Eq. (3.4), from the FH upper bound one has

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle\geq\int_{0}^{+\infty}\left\\{\bar{F}_{X}(t)+\bar{F}_{Y}(t)-2\min\\{\bar{F}_{X}(t),\bar{F}_{Y}(t)\\}\right\\}{\rm
d}t$ $\displaystyle=\int_{0}^{+\infty}|\bar{F}_{X}(t)-\bar{F}_{Y}(t)|{\rm
d}t,$

where the last equality follows from $|x-y|=x+y-2\min\\{x,y\\}$. Similarly,
from the FH lower bound we set

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle\leq\int_{0}^{+\infty}\left\\{\bar{F}_{X}(t)+\bar{F}_{Y}(t)-2\max\\{\bar{F}_{X}(t)+\bar{F}_{Y}(t)-1,0\\}\right\\}{\rm
d}t$
$\displaystyle=\int_{0}^{+\infty}\left\\{1-|\bar{F}_{X}(t)+\bar{F}_{Y}(t)-1|\right\\}{\rm
d}t,$

where the last equality follows from $2\max\\{x,y\\}=|x-y|+x+y$. This conclude
the proof. $\Box$

We remark that, if $X\leq_{st}Y$, then the lower bound in Eq. (4.4) coincides
with the one given in Eq. (4.2), attained when $C=M$.

###### Remark 4.2

Suppose that $X\sim\mathcal{U}(0,a)$ and $Y\sim\mathcal{U}(0,b)$, with
$0<a<b$. For any copula $C$, from Eq. (4.4) with some calculations it follows

$\frac{b^{2}-a^{2}}{2(a+b)}\leq\mathsf{GMD}(X,Y)\leq\frac{a^{2}+b^{2}}{2(a+b)},\qquad\frac{b^{2}-a^{2}}{(a+b)^{2}}\leq\mathsf{G}(X,Y)\leq\frac{a^{2}+b^{2}}{(a+b)^{2}}.$

Suppose now that $X\sim{\rm Exp}(1)$ and $Y\sim\mathcal{U}(0,1)$. Then from
Eq. (4.4) with some calculations one obtains

$0.5\leq\mathsf{GMD}(X,Y)\leq 0.955937,\qquad 0.333333\leq\mathsf{G}(X,Y)\leq
0.637291.$

In the next result we provide an upper bound for the bivariate Gini’s mean
difference when $X=_{st}Y$, with particular attention to the cases in which
the median of $X$ coincides with its mean value.

###### Proposition 4.4

Let $(X,Y)$ be a random vector with i.d. marginals having c.d.f. $F$, mean
$\mu$ and median $m$. Then

$0\leq\mathsf{GMD}(X,Y)\leq 2\left(\mu-m\right)+4\int_{0}^{m}F(t){\rm d}t.$
(4.5)

In particular, if $\mu=m$, then $\mathsf{GMD}(X,Y)\leq 4\int_{0}^{m}F(t){\rm
d}t$.

  * Proof.

Under these assumptions, from Eq. (4.4), with few calculations one has

$\displaystyle\mathsf{GMD}(X,Y)$
$\displaystyle\leq\int_{0}^{+\infty}1-|2\bar{F}(t)-1|{\rm d}t$
$\displaystyle=2\int_{0}^{m}F(t){\rm d}t+2\int_{m}^{+\infty}\bar{F}(t){\rm
d}t$ $\displaystyle=2\left(\mu-m\right)+4\int_{0}^{m}F(t){\rm d}t,$

where the last equality is obtained by adding and subtracting
$2\int_{0}^{m}\bar{F}(t){\rm d}t$. $\Box$

###### Remark 4.3

If $X$ and $Y$ are uniformly distributed over $[0,b]$, with $b>0$, from Eq.
(4.5), recalling that $\mu=m=1/2$, one has

$\mathsf{GMD}(X,Y)\leq\frac{b}{2},\qquad\mathsf{G}(X,Y)\leq\frac{1}{2}.$

If $X$ and $Y$ are exponentially distributed with mean $\mu>0$, from Eq. (4.5)
it follows

$\mathsf{GMD}(X,Y)\leq 2\mu\ln(2),\qquad\mathsf{G}(X,Y)\leq\ln(2).$

###### Remark 4.4

Making use of the Jensen’s inequality in Eq. (2.11), it is easy to see that
the median and the mean satisfy the following well known relation

$|\mu-m|\leq\sigma,$

where $\sigma$ is the standard deviation. Analogously, for an absolutely
continuous random lifetime $X$, from Markov’s inequality in Eq. (2.12) we get

${\rm Pr}(X\geq m)=\frac{1}{2}\leq\frac{\mu}{m}$

and therefore $\mu\geq m/2$. Moreover, from Eq. (4.5) one can obtain another
lower bound for the mean as

$\mu\geq m-2\int_{0}^{m}F(t){\rm d}t.$

In the next result we give bounds for the bivariate Gini’s mean difference
when $X$ and $Y$ are stochastically ordered, by using the bivariate Gini’s
mean differences of their respectively homogeneous cases. The bounds for the
bivariate Gini’s index follow dividing by $\mathsf{E}(X)+\mathsf{E}(Y)$.

###### Proposition 4.5

Let $(X,Y),(X,\tilde{X})$ and $(Y,\tilde{Y})$ be three random vectors with the
same survival copula $\widehat{C}$, where $X=_{st}\tilde{X}$ and
$Y=_{st}\tilde{Y}$, respectively. If $X\leq_{st}Y$, then

$\mathsf{GMD}(Y,\tilde{Y})-2\left(\mathsf{E}(Y)-\mathsf{E}(X)\right)\leq\mathsf{GMD}(X,Y)\leq\mathsf{GMD}(X,\tilde{X})+2\left(\mathsf{E}(Y)-\mathsf{E}(X)\right).$
(4.6)

  * Proof.

We only show how to obtain the right-hand-side of Eq. (4.6), since the left-
hand-side similarly follows. Under the stated assumptions, since
$\bar{F}_{X}\leq_{st}\bar{F}_{Y}$ one has $\mathsf{E}(X)\leq\mathsf{E}(Y)$ and
$\mathsf{E}(\min\\{X,\tilde{X}\\})\leq\mathsf{E}(\min\\{X,Y\\})$, and
therefore

$\mathsf{GMD}(X,Y)\leq 2\mathsf{E}(Y)-2\mathsf{E}(\min\\{X,\tilde{X}\\}).$

The proof follows making use of Eq. (3.3) in
$\mathsf{E}(\min\\{X,\tilde{X}\\})$. $\Box$

In the independent case, the Gini’s indices in the bounds given in Eq. (4.6)
can be replaced with the respectively univariate indices defined in Eq. (1.1).

##  5 Efficiency Gini’s indices in systems

In this section we define particular versions of the bivariate Gini’s indices,
for the cases in which the random lifetimes $X$ and $Y$ in Eqs. (3.1) and
(3.6) are suitable systems. First, we recall that a (binary) system with
(binary) components of order $n$ is a Boolean structure function (map)
$\Phi:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$, where
$\Phi\left(x_{1},\dots,x_{n}\right)\in\\{0,1\\}$ represents the system’s state
that is completely determined by the components’ states represented by
$x_{1},\dots,x_{n}\in\\{0,1\\}$. A semi-coherent system of order $n$ is a
system $\Phi$ that is increasing and such that $\Phi\left(0,\dots,0\right)=0$
and $\Phi\left(1,\dots,1\right)=1$. In addition, we say that $\Phi$ is a
coherent system of order $n$ if it is increasing and strictly increasing in
each variable in at least a point. We remark that this function $\Phi$ can be
extended to real numbers and then the random lifetime $T$ of the system can be
written as $T=\Phi(X_{1},\dots,X_{n})$, where $X_{1},\dots,X_{n}$ are the
random lifetimes of the components. For more details about systems see Navarro
[18].

We now are ready to define efficiency versions of the new Gini’s indices. The
purpose is to measure how good is a given system $T$.

###### Definition 5.1

Let $(X_{1},\dots,X_{n})$ be a random vector and let
$T=\Phi(X_{1},\dots,X_{k})$ be the lifetime of any semi-coherent system, for
$k\leq n$. Consider $X_{1:n}=\min\\{X_{1},\dots,X_{n}\\}$. The efficiency
Gini’s mean difference of order $n$ of $T$ is defined as

$\mathsf{GMD}_{n}(T)=\mathsf{GMD}(X_{1:n},T).$ (5.1)

Clearly, from Eq. (3.1), one has
$\mathsf{GMD}_{n}(T)=\mathsf{E}(T)-\mathsf{E}(X_{1:n})$. In addition,
$\mathsf{GMD}_{n}(X_{1:n})=0$ and, recalling Eq. (3.18),
$\mathsf{GMD}_{n}(X_{n:n})=\mathsf{GMD}({\bf X})$.

From Eq. (3.6), one can obtain a ratio based on Eq. (5.1) as
$\mathsf{GMD}_{n}(T)/(\mathsf{E}(T)+\mathsf{E}(X_{1:n}))$. However, in order
to give more information about the efficiency of a system, we prefer to define
the efficiency Gini’s index as

$\mathsf{G}_{n}(T)\coloneqq\frac{\mathsf{GMD}_{n}(T)}{\mathsf{GMD}_{n}(X_{n:n})},$
(5.2)

where $\mathsf{G}_{n}(T)=0$ if and only if $T\equiv X_{1:n}$, and
$\mathsf{G}_{n}(T)=1$ if and only if $T\equiv X_{n:n}$. Therefore, since Eq.
(5.1) represents the distance between the expected first component failure and
the expected failure of $T$, one can measure the efficiency of any semi-
coherent system by using Eq. (5.2). Hence, the efficiency of a system in terms
of its duration increases as $\mathsf{G}_{n}(T)$ increases. For predictions of
$T$ from $X_{1:n}$ see Navarro et al. [19].

Making use of Eq. (4.3) we provide below a useful interpretation of the
efficiency Gini’s index.

###### Proposition 5.1

Under the assumptions of Definition 5.1, for $c>0$ one has

$\mathsf{Pr}\left(T<X_{1:n}+c\,\mathsf{GMD}_{n}(X_{n:n})\right)\geq
1-\frac{\mathsf{G}_{n}(T)}{c}.$ (5.3)

  * Proof.

The thesis immediately follows from Eq. (4.3) when
$a=c\,\mathsf{GMD}_{n}(X_{n:n})$. $\Box$

Hence, Eq. (5.3) leads to consider the efficiency Gini’s index as a “control
index” for the probability that the system fails after the first failure in an
instant given by the difference between the expectations of the first and the
last component failures.

One could define a dual version of Eq. (5.1) for any semi-coherent system $T$,
referring to $X_{n:n}$ instead of $X_{1:n}$, as $\mathsf{GMD}(T,X_{n:n})$. In
this case the corresponding Gini’s index does not describe the same efficiency
pointed out in Eq. (5.2). Indeed, $\mathsf{GMD}(T,X_{n:n})$ measures the
distance between the last component failure and the failure of the system, and
therefore it represents a sort of “inefficiency”.

We now provide a signature representation of the efficiency Gini’s mean
difference defined in Eq. (5.1). Samaniego [30] (see also Samaniego [31])
introduced the first signature representation for the reliability of a
coherent system. The interpretation is intuitively justified by the fact that
a coherent system $T$ is going to fail with a component failure, and therefore
the reliability of $T$ can viewed as a weighted sum of the ordered statistics
(ordered component failures) $X_{i:n}$ for $i\in\\{1,\dots,n\\}$. Hence, if
$T$ has i.i.d. component lifetimes $X_{1},\dots,X_{n}$ with continuous s.f.
$\bar{F}$, then

$\bar{F}_{T}(t)=\sum_{i=1}^{n}s_{i}\bar{F}_{i:n}(t),$ (5.4)

for all $t$, where $s_{1},\dots,s_{n}$ are nonnegative coefficients such that
$\sum_{i=1}^{n}s_{i}=1$ and that do not depend on $\bar{F}$, and where
$\bar{F}_{i:n}$ is the reliability function of $X_{i:n}$, for
$i\in\\{1,\dots,n\\}$. The vector ${\bf s}=\left(s_{1},\dots,s_{n}\right)$ is
called the signature of the system, where
$s_{i}=\mathsf{Pr}\left(T=X_{i:n}\right)$ for $i\in\\{1,\dots,n\\}$. Recalling
that, for all $t$, the s.f. of $X_{i:n}$ is

$\bar{F}_{i:n}(t)=\sum_{j=0}^{i-1}\binom{n}{j}\left[F(t)\right]^{j}\left[\bar{F}(t)\right]^{n-j},$

(see, for instance, Proposition 2.2 in [18], p. 30), from Eq. (5.4), by
interchanging the order of summations, it follows

$\bar{F}_{T}(t)=\sum_{i=1}^{n}S_{n-i+1}\binom{n}{i}\left[F(t)\right]^{n-i}\left[\bar{F}(t)\right]^{i},$
(5.5)

where $S_{j}=\sum_{i=j}^{n}s_{i}$ for $j\in\\{1,\dots,n\\}$, is the
probability that the system works when exactly $n-j+1$ components work.

In Capaldo et al. [5] the authors defined the cumulative information
generating function of $X$ as

$G_{X}(\alpha,\beta)\coloneqq\int_{0}^{+\infty}\left[F(t)\right]^{\alpha}\left[\bar{F}(t)\right]^{\beta}\,{\rm{d}}t,$
(5.6)

for all $(\alpha,\beta)\in\mathbb{R}^{2}$ such that the right-hand-side of Eq.
(5.6) is finite. Recalling Eq. (1.1), it easy is to see that
$\mathsf{GMD}(X)=2G_{X}(1,1)$. See [5] also for different generalizations of
the univariate Gini’s mean difference in Eq. (1.1). Eq. (5.6) provides a
unifying mathematical tool suitable to deal with cumulative entropies based on
the s.f. and c.d.f., introduced and studied in Rao et al. [29] and Di
Crescenzo and Longobardi [11], respectively. In addition, it is able to
recover both generalized and fractional cumulative entropies, see Di Crescenzo
et al. [10], Kayal [16], Psarrakos and Navarro [28], Toomaj and Di Crescenzo
[33] and Xiong et al. [34] as references. Marginal versions of Eq. (5.6) have
been also discussed in [5]. In particular, we recall the cumulative residual
information generating measure of $X$, defined as

$K_{X}(\beta)\coloneqq G_{X}(0,\beta),$ (5.7)

for all $\beta\in\mathbb{R}$ such that $K_{X}(\beta)$ is finite.

In the next result we provide an alternative expression of Eq. (5.1) for $k=n$
in terms of Eq. (5.6), from the signature representation discussed above.

###### Theorem 5.1

Let $T=\Phi(X_{1},\dots,X_{n})$ be any coherent system having i.i.d.
components. Then

$\mathsf{GMD}_{n}(T)=\sum_{i=1}^{n-1}S_{n-i+1}\binom{n}{i}G_{X}(n-i,i).$

  * Proof.

Making use of Eq. (5.5) for the calculation of $\mathsf{E}(T)$ in Eq. (5.1),
the thesis immediately follows from Eq. (5.6). $\Box$

The Samaniego’s representation does not necessarily hold in the general case.
Another way to compute the system reliability from a signature representation
is given in Navarro et al. [20], namely minimal signature representation,
making use of the concept of minimal path set representation (for further
details see Section 2.3 in [18]). Therefore, if $T$ is the lifetime of a
coherent (or semi-coherent) system with exc. component lifetimes
$X_{1},\dots,X_{n}$, then

$\bar{F}_{T}(t)=\sum_{i=1}^{n}a_{i}\bar{F}_{1:i}(t),$ (5.8)

for all $t$, where $a_{1},\dots,a_{n}$ are some integer coefficients such that
$a_{1}+\cdots+a_{n}=1$, and
$\bar{F}_{1:i}(t)=\mathsf{Pr}\left(\min\\{X_{1},\dots,X_{i}\\}>t\right)$ for
$i\in\\{1,\dots,n\\}$. The vector ${\bf a}=\left(a_{1},\dots,a_{n}\right)$ is
called the minimal signature of $T$. Similarly to Theorem 5.1, in the next
result we provide an alternative expression of Eq. (5.1) in terms of Eqs.
(5.7) and (5.8).

###### Theorem 5.2

Let $T=\Phi(X_{1},\dots,X_{n})$ be any coherent system having i.i.d.
components. Then

$\mathsf{GMD}_{n}(T)=\sum_{i=1}^{n-1}a_{i}K_{X}(i)+(a_{n}-1)K_{X}(n).$

  * Proof.

By using Eq. (5.8) in the calculation of $\mathsf{E}(T)$ in Eq. (5.1), the
thesis immediately follows from Eq. (5.7). $\Box$

The Samaniego’s representation can also be extended to semi-coherent systems,
making use of the structural signature ${\bf s}^{(n)}$ of order $n$. A result
similar to Theorem 5.1 holds for ${\bf s}^{(n)}$. Analogously, ${\bf
a}^{(n)}=(a_{1},\dots,a_{k},0,\dots,0)$ is called the minimal signature of
order n (see [18], pp. 48-49). Along the same line of Theorem 5.2, below we
provide an alternative expression of Eq. (5.1) in terms of Eq. (5.7), by using
the minimal signature of order $n$.

###### Theorem 5.3

Let $T=\Phi(X_{1},\dots,X_{k})$ be any semi-coherent system having i.i.d.
components, contained in a random vector $(X_{1},\dots,X_{n})$. Therefore

$\mathsf{GMD}_{n}(T)=\sum_{i=1}^{k}a_{i}K_{X}(i)-K_{X}(n).$ (5.9)

Consider now an exc. random vector $(X_{1},\dots,X_{n})$ having marginals with
common s.f. $\bar{F}$. Then for any semi-coherent system
$T=\Phi(X_{1},\dots,X_{k})$, for $k\leq n$, one has

$\mathsf{GMD}_{n}(T)=\sum_{i=1}^{k}a_{i}\int_{0}^{+\infty}\widehat{\delta}_{i}(\bar{F}(t)){\rm
d}t-\int_{0}^{+\infty}\widehat{\delta}_{n}(\bar{F}(t)){\rm d}t,$ (5.10)

where $\widehat{\delta}_{i}(u)=\widehat{C}(u,\dots,u,1,\dots,1)$, for $u$
repeated $i$-times, is the diagonal section of the copula of
$(X_{1},\dots,X_{i})$, for $i\in\\{1,\dots,n\\}$ and $u\in[0,1]$. Clearly, in
the i.i.d. case Eq. (5.10) reduces to Eq. (5.9).

In Table 1, making use of Eq. (5.9), we compute the efficiency Gini’s index
for all the coherent systems with $1$-$4$ i.i.d. components, uniformly or
exponentially distributed. Table 2 shows the efficiency Gini’s index for all
coherent systems with $1$-$4$ i.d. components, uniformly or exponentially
distributed, computed using Eq. (5.10) under the following $4$-dimensional FGM
copula

$\widehat{C}(u_{1},u_{2},u_{3},u_{4})=u_{1}u_{2}u_{3}u_{4}\left(1+\theta(1-u_{1})(1-u_{2})(1-u_{3})(1-u_{4})\right),\qquad
u_{1},u_{2},u_{3},u_{4}\in[0,1],$ (5.11)

with diagonal section

$\widehat{\delta}_{4}(u)=u^{4}\left(1+\theta(1-u)^{4}\right),\qquad
u\in[0,1],$

for $\theta=1,-1$, where $\widehat{\delta}_{i}(u)=u^{i}$, for $i=1,2,3$.

Finally, we remark that Eq. (5.10) can also be extended to the not exc. case,
making use of minimal path set representation for semi-coherent systems (see
[18], p. 37).

Table 1: Efficiency Gini’s index for all coherent systems with $1$-$4$ i.i.d.
components, uniformly or exponentially distributed, making use of the minimal
signature representation of order $4$. The values have been calculated from
Eq. (5.9), by rounding to the third decimal place.

i | $T_{i}$ | ${\bf a^{(4)}}$ | $\mathsf{G}_{n}(T)$ | $\mathsf{G}_{n}(T)$
---|---|---|---|---
|  |  | $X_{j}\sim\mathcal{U}(0,1)$ | $X_{j}\sim{\rm Exp}(1)$
1 | $X_{1:1}=X_{1}$ | $(1,0,0,0)$ | $0.500$ | $0.409$
2 |  $X_{1:2}=\min(X_{1},X_{2})$ (2-series) | $(0,1,0,0)$ | $0.222$ | $0.136$
3 |  $X_{2:2}=\max(X_{1},X_{2})$ (2-parallel) | $(2,-1,0,0)$ | $0.778$ | $0.682$
4 |  $X_{1:3}=\min(X_{1},X_{2},X_{3})$ (3-series) | $(0,0,1,0)$ | $0.083$ | $0.045$
5 | $\min(X_{1},\max(X_{2},X_{3}))$ | $\left(0,2,-1,0\right)$ | $0.361$ | $0.227$
6 |  $X_{2:3}$ (2-out-of-3) | $(0,3,-2,0)$ | $0.500$ | $0.318$
7 | $\max(X_{1},\min(X_{2},X_{3}))$ | $\left(1,1,-1,0\right)$ | $0.639$ | $0.500$
8 |  $X_{3:3}=\max(X_{1},X_{2},X_{3})$ (3-parallel) | $(3,-3,1,0)$ | $0.917$ | $0.864$
9 |  $X_{1:4}=\min(X_{1},X_{2},X_{3},X_{4})$ (4-series) | $(0,0,0,1)$ | $0$ | $0$
10 | $\max(\min(X_{1},X_{2},X_{3}),\min(X_{2},X_{3},X_{4}))$ | $\left(0,0,2,-1\right)$ | $0.167$ | $0.091$
11 | $\min(X_{2:3},X_{4})$ | $\left(0,0,3,-2\right)$ | $0.250$ | $0.136$
12 | $\min(X_{1},\max(X_{2},X_{3}),\max(X_{3},X_{4}))$ | $\left(0,1,1,-1\right)$ | $0.306$ | $0.182$
13 | $\min(X_{1},\max(X_{2},X_{3},X_{4}))$ | $\left(0,3,-3,1\right)$ | $0.417$ | $0.273$
14 |  $X_{2:4}$ (3-out-of-4) | $(0,0,4,-3)$ | $0.333$ | $0.182$
15 | $\max(\min(X_{1},X_{2}),\min(X_{1},X_{3},X_{4}),\min(X_{2},X_{3},X_{4}))$ | $(0,1,2,-2)$ | $0.389$ | $0.227$
16 | $\max(\min(X_{1},X_{2}),\min(X_{3},X_{4}))$ | $(0,2,0,-1)$ | $0.444$ | $0.273$
17 | $\max(\min(X_{1},X_{2}),\min(X_{1},X_{3}),\min(X_{2},X_{3},X_{4}))$ | $(0,2,0,-1)$ | $0.444$ | $0.273$
18 | $\max(\min(X_{1},X_{2}),\min(X_{2},X_{3}),\min(X_{3},X_{4}))$ | $\left(0,3,-2,0\right)$ | $0.500$ | $0.318$
19 | $\max(\min(X_{1},\max(X_{2},X_{3},X_{4})),\min(X_{2},X_{3},X_{4}))$ | $\left(0,3,-2,0\right)$ | $0.500$ | $0.318$
20 | $\min(\max(X_{1},X_{2}),\max(X_{1},X_{3}),\max(X_{2},X_{3},X_{4}))$ | $\left(0,4,-4,1\right)$ | $0.556$ | $0.364$
21 | $\min(\max(X_{1},X_{2}),\max(X_{3},X_{4}))$ | $\left(0,4,-4,1\right)$ | $0.556$ | $0.364$
22 | $\min(\max(X_{1},X_{2}),\max(X_{1},X_{3},X_{4}),\max(X_{2},X_{3},X_{4}))$ | $\left(0,5,-6,2\right)$ | $0.611$ | $0.409$
23 |  $X_{3:4}$ (2-out-of-4) | $\left(0,6,-8,3\right)$ | $0.667$ | $0.455$
24 | $\max(X_{1},\min(X_{2},X_{3},X_{4}))$ | $\left(1,0,1,-1\right)$ | $0.583$ | $0.455$
25 | $\max(X_{1},\min(X_{2},X_{3}),\min(X_{3},X_{4}))$ | $\left(1,2,-3,1\right)$ | $0.694$ | $0.545$
26 | $\max(X_{2:3},X_{4})$ | $\left(1,3,-5,2\right)$ | $0.750$ | $0.591$
27 | $\min(\max(X_{1},X_{2},X_{3}),\max(X_{2},X_{3},X_{4}))$ | $\left(2,0,-2,1\right)$ | $0.833$ | $0.727$
28 |  $X_{4:4}=\max(X_{1},X_{2},X_{3},X_{4})$ (4-parallel) | $(4,-6,4,-1)$ | $1$ | $1$

Table 2: Efficiency Gini’s index for all coherent systems considered in Table
1, in this case with $1$-$4$ i.d. components, uniformly or exponentially
distributed, having FGM copula in Eq. (5.11), for $\theta=1,-1$ and making use
of the minimal signature representation of order $4$. The values have been
calculated from Eq. (5.10), by rounding to the third decimal place.

i | ${\bf a^{(4)}}$ | $\mathsf{G}_{n}(T)$ | $\mathsf{G}_{n}(T)$ | $\mathsf{G}_{n}(T)$ | $\mathsf{G}_{n}(T)$
---|---|---|---|---|---
|  | $X_{j}\sim\mathcal{U}(0,1),\theta=1$ | $X_{j}\sim\mathcal{U}(0,1),\theta=-1$ | $X_{j}\sim{\rm Exp}(1),\theta=1$ | $X_{j}\sim{\rm Exp}(1),\theta=-1$
1 | $(1,0,0,0)$ | $0.500$ | $0.500$ | $0.409$ | $0.409$
2 | $(0,1,0,0)$ | $0.221$ | $0.224$ | $0.135$ | $0.138$
3 | $(2,-1,0,0)$ | $0.779$ | $0.776$ | $0.683$ | $0.681$
4 | $(0,0,1,0)$ | $0.081$ | $0.086$ | $0.044$ | $0.047$
5 | $\left(0,2,-1,0\right)$ | $0.360$ | $0.362$ | $0.226$ | $0.228$
6 | $(0,3,-2,0)$ | $0.500$ | $0.500$ | $0.317$ | $0.319$
7 | $\left(1,1,-1,0\right)$ | $0.640$ | $0.638$ | $0.500$ | $0.500$
8 | $(3,-3,1,0)$ | $0.919$ | $0.914$ | $0.865$ | $0.862$
9 | $(0,0,0,1)$ | $0$ | $0$ | $0$ | $0$
10 | $\left(0,0,2,-1\right)$ | $0.162$ | $0.171$ | $0.087$ | $0.094$
11 | $\left(0,0,3,-2\right)$ | $0.243$ | $0.257$ | $0.131$ | $0.142$
12 | $\left(0,1,1,-1\right)$ | $0.302$ | $0.309$ | $0.179$ | $0.185$
13 | $\left(0,3,-3,1\right)$ | $0.419$ | $0.414$ | $0.274$ | $0.272$
14 | $(0,0,4,-3)$ | $0.324$ | $0.342$ | $0.175$ | $0.189$
15 | $(0,1,2,-2)$ | $0.383$ | $0.395$ | $0.222$ | $0.232$
16 | $(0,2,0,-1)$ | $0.441$ | $0.447$ | $0.270$ | $0.276$
17 | $(0,2,0,-1)$ | $0.441$ | $0.447$ | $0.270$ | $0.276$
18 | $\left(0,3,-2,0\right)$ | $0.500$ | $0.500$ | $0.317$ | $0.319$
19 | $\left(0,3,-2,0\right)$ | $0.500$ | $0.500$ | $0.317$ | $0.319$
20 | $\left(0,4,-4,1\right)$ | $0.559$ | $0.553$ | $0.365$ | $0.362$
21 | $\left(0,4,-4,1\right)$ | $0.559$ | $0.553$ | $0.365$ | $0.362$
22 | $\left(0,5,-6,2\right)$ | $0.617$ | $0.605$ | $0.413$ | $0.406$
23 | $\left(0,6,-8,3\right)$ | $0.676$ | $0.658$ | $0.460$ | $0.449$
24 | $\left(1,0,1,-1\right)$ | $0.581$ | $0.586$ | $0.452$ | $0.457$
25 | $\left(1,2,-3,1\right)$ | $0.698$ | $0.691$ | $0.548$ | $0.543$
26 | $\left(1,3,-5,2\right)$ | $0.757$ | $0.743$ | $0.595$ | $0.587$
27 | $\left(2,0,-2,1\right)$ | $0.838$ | $0.829$ | $0.730$ | $0.724$
28 | $(4,-6,4,-1)$ | $1$ | $1$ | $1$ | $1$

##  6 Examples and simulations

In this section we show examples of the bivariate Gini’s indices defined in
Section 3. Interpretations in terms of areas also arises for the bivariate
Gini’s index in Eq. (3.6), as for the univariate one given in Eq. (1.2). We
also define empirical Gini’s indices to approximate them from data, by
providing simulations. We also compute the efficiency Gini’s index for some
systems.

In the first example we consider bivariate Gini’s indices of $(X,Y)$ with i.d.
marginals, according to the uniform distribution, under different copulas.
Similarly to the univariate case, the bivariate Gini’s index turns out to be
two times the area between the egalitarian line and the diagonal section of
$(X,Y)$.

###### Example 6.1

Let $(X,Y)$ be a random vector with diagonal section $\delta$. If $X$ and $Y$
are uniformly distributed over $[0,b]$, with $b>0$, from Eq. (3.5) it follows
that

$\mathsf{GMD}(X,Y)=2b\int_{0}^{1}\left\\{t-\delta(t)\right\\}{\rm
d}t=b\left(1-2\int_{0}^{1}\delta(t){\rm d}t\right)$

and, from Eq. (3.6), one has

$\mathsf{G}(X,Y)=2\int_{0}^{1}\left\\{t-\delta(t)\right\\}{\rm
d}t=1-2\int_{0}^{1}\delta(t){\rm d}t,$ (6.1)

that does not depend on $b$. The first expression in Eq. (6.1) is similar to
the representation for the univariate Gini’s index as two times the area
between the Lorenz curve and the line $y=x$ (see Fig. 5.2 in [4]). For
example, referring to Eq. (6.1), in the left-hand-side of Fig. 1 we plot the
area (dark grey) in the i.i.d. case, i.e., for $\delta(t)=t^{2}$ for
$t\in[0,1]$ (black line). We obtain an area of $1/6$ and so
$\mathsf{G}(X,Y)=1/3$, that clearly represents also the value of the
univariate Gini’s index of the uniform distribution (since $X$ and $Y$ are
independent). We also add the two extreme cases obtained from the FH bounds in
Eq. (2.4). For $M$ (red line) we get the lower bound for the index
$\mathsf{G}(X,Y)=0$, since $\delta(t)=t$. For $W$ (blue line) we obtain an
area of $1/4$ (dark and light grey) and therefore $\mathsf{G}(X,Y)=1/2$ is the
upper bound for the Gini’s index of any bidimensional copula with uniform
marginals (as shown in Remark 4.3). Moreover, referring to Eq. (6.1), in the
right-hand-side of Fig. 1 one can see the areas for the diagonal section of
the FGM family of copulas defined in Eq. (2.5) for $\theta=0$ (black, i.i.d.
case), $1,0.5$ (orange) and $-1,-0.5$ (green). We provide the areas for
$\theta=1,-1$ $(2/15,3/15)$ that lead to the extreme Gini’s indices
$\mathsf{G}(X,Y)=4/15=0.2666667$ and $\mathsf{G}(X,Y)=2/5=0.4$ for this family
of copulas. Note that the case $\theta=0$ gives the bound
$\mathsf{G}(X,Y)=1/3$ for positive $(\theta>0)$ and negative dependence cases
$(\theta<0)$. In this family the Gini’s index is decreasing with $\theta$, as
expected.

Similarly, referring to Eq. (6.1), in the left-hand-side of Fig. 2 we plot
$\delta$ for the Clayton copulas defined in Eq. (2.6) with
$\theta=1,2,5,10,20$ (orange) and $\theta=-0.2,-0.4,-0.6,-0.8$ (green). The
independent case is obtained when $\theta\to 0$ (black) while $M$ is obtained
when $\theta\to\infty$ (red) and $W$ for $\theta=-1$ (blue). We provide the
areas for $\theta=1$ $(0.1137056)$ and $\theta=-0.8$ $(0.2340531)$ that lead
to $\mathsf{G}(X,Y)=0.2274112$ and $\mathsf{G}(X,Y)=0.4681062$, respectively.
The upper bound is $0.5$ (blue line) and $1/3$ is also a bound for positive
$(\theta>0)$ and negative $(\theta<0)$ dependence cases. Moreover, referring
to Eq. (6.1), in the right-hand-side of Fig. 2 we plot $\delta$ for the Frank
copulas defined in Eq. (2.7) with $\theta=1,2,5,10,20$ (orange) and
$\theta=-1,-2,-5,-10,-20$ (green). The independent case is obtained when
$\theta\to 0$ (black) while $M$ is obtained when $\theta\to+\infty$ (red) and
$W$ for $\theta\to-\infty$ (blue). We provide the areas for $\theta=1$
$(0.1498039)$ and $\theta=-1$ $(0.1827476)$ that lead to
$\mathsf{G}(X,Y)=0.2996078$ and $\mathsf{G}(X,Y)=0.3654952$, respectively.

Fig. 1: Plots and areas referring to Eq. (6.1) for the diagonal section
$\delta$ in the independent case (black), $M$ (red) and $W$ (blue) and FGM
copulas for $\theta=1,0.5$ (orange) and $-1,-0.5$ (green).

Fig. 2: Plots and areas referring to Eq. (6.1) for the diagonal section
$\delta$ of Clayton copulas (left-hand-side) for $\theta=1,2,5,10,20$ (orange)
and $\theta=-0.2,-0.4,-0.6,-0.8$ (green), while Frank copulas (right-hand-
side) for $\theta=1,2,5,10,20$ (orange) and $\theta=-1,-2,-5,-10,-20$ (green).

In the second example we consider bivariate Gini’s indices of $(X,Y)$ with
i.d. marginals, according to the exponential distribution, under different
copulas. In this model, the interpretation in terms of area is different from
the univariate case. Indeed, the bivariate Gini’s index turns out to be the
area between the unitary line and the ratio based on the diagonal section of
$(X,Y)$ and the egalitarian line.

###### Example 6.2

Let $(X,Y)$ be a random vector with survival diagonal section
$\widehat{\delta}$. If $X$ and $Y$ are exponentially distributed with mean
$\mu$, from Eq. (3.1) it follows that

$\mathsf{GMD}(X,Y)=2\mu\int_{0}^{1}\left\\{1-\frac{\hat{\delta}(t)}{t}\right\\}{\rm
d}t$

and, from Eq. (3.6), one has

$\mathsf{G}(X,Y)=1-\int_{0}^{1}\frac{\hat{\delta}(t)}{t}{\rm d}t.$ (6.2)

Eq. (6.2) just depends on the diagonal section
$\widehat{\delta}(u)=\widehat{C}(u,u)$, for all $u\in[0,1]$, and the bivariate
Gini’s index can also be represented as the area between $1$ and
$\hat{\delta}(u)/u$. It is easy to see that in the PQD (NQD) case one has
$\widehat{\delta}(u)\geq u^{2}$ $(\leq)$ and $\mathsf{G}(X,Y)\leq 1/2$
$(\geq)$. For example, in the left-hand-side of Fig. 3 we plot the area (dark
grey) in the i.i.d. case, i.e., $\hat{\delta}(t)=t^{2}$ for $t\in[0,1]$ (black
line), and we also add the two extreme cases obtained from the FH bounds in
Eq. (2.4). For the i.i.d. case we obtain an area of $1/2$ and therefore
$\mathsf{G}(X,Y)=1/2$, that is also the value of the univariate Gini’s index
of the exponential distribution (since $X$ and $Y$ are independent). For $M$
(red line) we get the lower bound for the index $\mathsf{G}(X,Y)=0$, while for
$W$ (blue line) we obtain an area of $\ln 2$ (dark and light grey) and
therefore $\mathsf{G}(X,Y)=\ln 2=0.6931472$ is the upper bound for the Gini’s
index of any bidimensional copula with exponential marginal distributions (as
shown in Remark 4.3). Moreover, in the right-hand-side of Fig. 3 one can see
the areas for the FGM family of copulas defined in Eq. (2.5) with $\theta=0$
(black, i.i.d. case), $1,0.5$ (orange) and $-1,-0.5$ (green). The respective
Gini’s indices are $\mathsf{G}(X,Y)=0.4166667$ and
$\mathsf{G}(X,Y)=0.5833333$.

Fig. 3: Plots and areas referring to Eq. (6.2) for $\hat{\delta}(u)/u$ in the
exponential independent case (black), $M$ (red) and $W$ (blue) and FGM copulas
for $\theta=1,0.5$ (orange) and $-1,-0.5$ (green).

By recalling Eqs. (3.2) and (3.7) we now define empirical versions of the
bivariate Gini’s indices introduced in Eqs. (3.1) and (3.6), respectively.

###### Definition 6.1

Let $(X,Y)$ be a random vector. Let $(X_{i},Y_{i})$ be i.i.d. from $(X,Y)$,
for $i\in\\{1,\dots,n\\}$. Consider $L_{i}=\min(X_{i},Y_{i})$ and
$U_{i}=\max(X_{i},Y_{i})$. The empirical bivariate Gini’s mean difference and
empirical bivariate Gini’s index of $(X,Y)$ are defined respectively as

$\mathsf{\widehat{GMD}}(X,Y)=\bar{U}-\bar{L},\qquad\mathsf{\widehat{G}}(X,Y)=\frac{\bar{U}-\bar{L}}{\bar{U}+\bar{L}},$

where $\bar{L}=\frac{1}{n}\sum_{i=1}^{n}L_{i}$ and
$\bar{U}=\frac{1}{n}\sum_{i=1}^{n}U_{i}$.

Note that $Z_{i}=|X_{i}-Y_{i}|$, for $i\in\\{1,\dots,n\\}$, are i.i.d. random
variables from $Z=|X-Y|$ and that

$\mathsf{\widehat{GMD}}(X,Y)=\frac{1}{n}\sum_{i=1}^{n}Z_{i}=\bar{Z},\qquad\mathsf{\widehat{G}}(X,Y)=\frac{\sum_{i=1}^{n}Z_{i}}{\sum_{i=1}^{n}(X_{i}+Y_{i})}=\frac{\bar{Z}}{\bar{X}+\bar{Y}}$

and therefore we can apply here all the classic convergence theorems for the
sample means. Let us see two examples.

###### Example 6.3

To get a simulated sample we consider the Clayton copula in Eq. (2.6) for
$\theta=1$ given by

$C(u,v)=\frac{uv}{u+v-uv},\qquad u,v\in[0,1].$

If $X=_{st}Y\sim\mathcal{U}(0,1)$, then the conditional distribution function
of $Y|X=u$ for $u\in(0,1)$ is

$C_{2|1}(v|u)=\partial_{1}C(u,v)=\frac{v^{2}}{(u+v-uv)^{2}},\qquad v\in[0,1].$

In order to simulate $Y|X=u$ from a given value $u$, we obtain the inverse
function of $C_{2|1}$ as

$C^{-1}_{2|1}(z|u)=\frac{u\sqrt{z}}{1-(1-u)\sqrt{z}},\qquad z\in[0,1].$ (6.3)

To get a sample with $100$ data we simulate $X_{i}\sim\mathcal{U}(0,1)$ for
$i\in\\{1,\dots,100\\}$ and then we obtain $Y_{i}$ from Eq. (6.3) with
$u=X_{i}$ and $z=Z_{i}\sim\mathcal{U}(0,1)$ for $i\in\\{1,\dots,100\\}$.
Therefore we get the estimation for the bivariate Gini’s index
$\widehat{\mathsf{G}}(X,Y)=0.2357458$ given in Definition 6.1. The exact value
is $\mathsf{G}(X,Y)=2\cdot 0.1137056=0.2274112$ (see Fig. 2, left). The data
obtained for $(L_{i},U_{i})$ can be seen in the left-hand-side of Fig. 4. To
get a sample from standard exponential distributions if
$\tilde{X}=_{st}\tilde{Y}\sim\rm{Exp}(1)$ with this survival copula we just
apply the inverse transform $\bar{F}^{-1}(z)=-\ln(z)$ to the above uniform
data obtaining the estimation $\widehat{\mathsf{G}}(X,Y)=0.5\cdot
0.6366323=0.3183161$. The exact value is $\mathsf{G}(X,Y)=0.3068528$. The data
obtained for $(\tilde{L}_{i},\tilde{U}_{i})$ can be seen in the right-hand-
side of Fig. 4.

Fig. 4: Simulated ordered data obtained from the Clayton copula in Example 6.3
with standard uniform (left-hand-side) or exponential (right-hand-side)
distributions.

###### Example 6.4

We now consider the Frank copula in Eq. (2.7) for $\theta=-1$ given by

$C(u,v)=\ln\left(1+\frac{(e^{u}-1)(e^{v}-1)}{e-1}\right),\qquad u,v\in[0,1].$

If $X=_{st}Y\sim\mathcal{U}(0,1)$, then the conditional distribution function
of $Y|X=u$ for $u\in(0,1)$ is

$C_{2|1}(v|u)=\frac{e^{u}(e^{v}-1)}{e-e^{u}+e^{v}(e^{u}-1)},\qquad v\in[0,1]$

and thus the inverse function of $C_{2|1}$ is

$C^{-1}_{2|1}(z|u)=\ln\left[\frac{(e^{u}-e)z-e^{u}}{(e^{u}-1)z-e^{u}}\right],\qquad
z\in[0,1].$ (6.4)

As before, from a sample with $100$ data we get the estimation for the
bivariate Gini’s index $\widehat{\mathsf{G}}(X,Y)=0.377477$ given in
Definition 6.1 by using Eq. (6.4). The exact value is $\mathsf{G}(X,Y)=2\cdot
0.1827476=0.3654952$ (see Fig. 2, right). The data obtained for
$(L_{i},U_{i})$ can be seen in the left-hand-side of Fig. 5. To get a sample
from standard exponential distributions if
$\tilde{X}=_{st}\tilde{Y}\sim\rm{Exp}(1)$ with this survival copula we just
apply the inverse transform $\bar{F}^{-1}(z)=-\ln(z)$ to the above uniform
data obtaining the estimation $\widehat{\mathsf{G}}(X,Y)=0.5\cdot
1.069694=0.534847$. The exact value is $\mathsf{G}(X,Y)=0.539814$. The data
obtained for $(\tilde{L}_{i},\tilde{U}_{i})$ can be seen in the right-hand-
side of Fig. 5.

In the last example we compute the empirical efficiency Gini’s index of two
coherent systems.

###### Example 6.5

Consider the system $T_{16}$ given in Table 1, with i.i.d. components having
standard exponential distributions. From a sample with $100$ data we get the
estimation of the efficiency Gini’s index $\widehat{\mathsf{G}}_{4}(T)=0.250$.
The exact value is $\mathsf{G}_{4}(T)=0.273$ as shown in Table 1. The data
obtained for $(X_{1:4},T)$ can be seen in the left-hand-side of Fig. 6.
Similarly, if one consider the system $T_{22}$ given in Table 1, with i.i.d.
components having standard exponential distribution, then
$\widehat{\mathsf{G}}_{4}(T)=0.416$. The exact value is
$\mathsf{G}_{4}(T)=0.409$ as shown in Table 1. The data obtained for
$(X_{1:4},T)$ can be seen in the right-hand-side of Fig. 6.

Fig. 5: Simulated ordered data obtained from the Frank copula in Example 6.4
with standard uniform (left-hand-side) or exponential (right-hand-side)
distributions.

Fig. 6: Simulated ordered data obtained of $(X_{1:4},T)$, for the systems
$T_{16}$ (left-hand-side) and $T_{22}$ (right-hand-side) introduced in Table
1, with i.i.d. components having standard exponential distributions.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

M.C. is member of the group GNCS of INdAM (Istituto Nazionale di Alta
Matematica). This work is partially supported by MUR-PRIN 2022, project
2022XZSAFN “Anomalous Phenomena on Regular and Irregular Domains:
Approximating Complexity for the Applied Sciences”, and MUR-PRIN 2022 PNRR,
project P2022XSF5H “Stochastic Models in Biomathematics and Applications”.
M.C. expresses his warmest thanks to the Departamento de Estadística e
Investigación Operativa of Universidad de Murcia for the hospitality during a
three-month visit carried out in 2023. J.N. thanks the support of Ministerio
de Ciencia e Innovación of Spain under grants
PID2019-103971GB-I00/AEI/10.13039/501100011033 and
MCIN/AEI/10.13039/501100011033 and the project TED2021-129813A-I00 with the
support of the European Union “NextGenerationEU”/PRTR.

## References

  * [1]
  * [2] Arnold, B.C., Castillo, E., Sarabia, J.M., 1999. Conditional Specification of Statistical Models. New York, NY: Springer New York.
  * [3] Arnold, B.C., Castillo, E., Sarabia, J.M., 2012. Conditionally Specified Distributions. Springer Science & Business Media.
  * [4] Arnold, B.C., Sarabia, J.M., 2018. Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics. Statistics for Social and Behavioral Sciences. Springer.
  * [5] Capaldo, M., Di Crescenzo, A., Meoli, A., 2023. Cumulative information generating function and generalized Gini functions. Metrika: http://dx.doi.org/10.1007/s00184-023-00931-3.
  * [6] Capaldo, M., Di Crescenzo, A., Pellerey, F., 2024. Generalized Gini’s mean difference through distortions and copulas, and related minimizing problems. Statistics & Probability Letters, 206, 109981.
  * [7] Caramellino, L., Spizzichino, F., 1994. Dependence and aging properties of lifetimes with Schur-constant survival functions. Probability in the Engineering and Informational Sciences 8(1), 103-111.
  * [8] Ceriani, L., Verme, P., 2012. The origins of the Gini index: extracts from Variabilità e Mutabilità (1912) by Corrado Gini. The Journal of Economic Inequality 10(3), 421-443.
  * [9] Di Crescenzo, A., 2007. A Parrondo paradox in reliability theory. The Mathematical Scientist 32(1), 17–22.
  * [10] Di Crescenzo, A., Kayal, S., Meoli, A., 2021. Fractional generalized cumulative entropy and its dynamic version. Communications in Nonlinear Science and Numerical Simulation 102, 105899.
  * [11] Di Crescenzo, A., Longobardi, M., 2009. On cumulative entropies. Journal of Statistical Planning and Inference 139(12), 4072-4087.
  * [12] Durante, F., Papini, P.L., 2007. A weakening of Schur-concavity for copulas. Fuzzy Sets and Systems 158(12), 1378-1383.
  * [13] Durante, F., Sempi, C., 2016. Principles of Copula Theory. CRC/Chapman & Hall, London.
  * [14] Fuchs, S., Tschimpke, M., 2024. A novel positive dependence property and its impact on a popular class of concordance measures. Journal of Multivariate Analysis 200, 105259.
  * [15] Gastwirth, J.L., 1972. The estimation of the Lorenz curve and Gini index. The review of economics and statistics, 306-316.
  * [16] Kayal, S., 2016. On generalized cumulative entropies. Probability in the Engineering and Informational Sciences 30(4), 640-662.
  * [17] Koshevoy, G.A., Mosler, K., 1997. Multivariate gini indices. Journal of Multivariate Analysis 60(2), 252-276.
  * [18] Navarro, J., 2022. Introduction to System Reliability Theory. Springer.
  * [19] Navarro, J., Arriaza, A., Suárez-Llorens, A., 2023. Predicting failure times of coherent systems. To appear in Naval Research and Logistics.
  * [20] Navarro, J., Ruiz, J.M., Sandoval, C.J., 2007. Properties of coherent systems with dependent components. Communications in Statistics Theory and Methods 36(1), 175–191.
  * [21] Navarro, J., Sordo, M.A., 2018. Stochastic comparisons and bounds for conditional distributions by using copula properties. Dependence Modeling 6(1), 156-177.
  * [22] Navarro, J., Spizzichino, F., 2010. Comparisons of series and parallel systems with components sharing the same copula. Applied Stochastic Models in Business and Industry 26(6), 775-791.
  * [23] Nelsen, R.B., 2005. Some properties of Schur-constant survival models and their copulas. Brazilian Journal of Probability and Statistics 19(2), 179-190.
  * [24] Nelsen, R.B., 2006. An Introduction to Copulas. Springer Science & Business Media.
  * [25] Ortega-Jiménez, P., Pellerey, F., Sordo, M.A., Suárez-Llorens, A., 2022. A minimizing problem of distances between random variables with proportional reversed hazard rate functions. In: Building Bridges Between Soft and Statistical Methodologies for Data Science. Springer, Cham, pp. 311-318.
  * [26] Ortega-Jiménez, P., Sordo, M.A., Suárez-Llorens, A., 2021. Stochastic comparisons of some distances between random variables. Mathematics 9(9), 981.
  * [27] Pellerey, F., Navarro, J., 2022. Stochastic monotonicity of dependent variables given their sum. Test 31(2), 543-561.
  * [28] Psarrakos, G., Navarro, J., 2013. Generalized cumulative residual entropy and record values. Metrika 76(5), 623-640.
  * [29] Rao, M., Chen, Y., Vemuri, B.C., Wang, F., 2004. Cumulative residual entropy, a new measure of information. IEEE Transactions on Information Theory 50(6), 1220-1228.
  * [30] Samaniego, F.J., 1985. On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability R-34(1), 69–72.
  * [31] Samaniego, F.J., 2007. System Signatures and Their Applications in Engineering Reliability. International Series in Operations Research & Management Science 110, Springer, New York.
  * [32] Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer Series in Statistics. Springer, New York.
  * [33] Toomaj, A., Di Crescenzo, A., 2020. Generalized entropies, variance and applications. Entropy 22(6), 709.
  * [34] Xiong, H., Shang, P., Zhang, Y., 2019. Fractional cumulative residual entropy. Communications in Nonlinear Science and Numerical Simulation 78, 104879.
  * [35] Yin, X., Balakrishnan, N., Yin, C., 2023. Bounds for Gini’s mean difference based on first four moments, with some applications. Statistical Papers 64, 2081-2100.
  * [36] Yitzhaki, S., 2003. Gini’s mean difference: a superior measure of variability for non-normal distributions. Metron 61(2), 285-316.
  * [37]