#  Cumulative Information Generating Function and Generalized Gini Functions

Marco Capaldo Antonio Di Crescenzo Alessandra Meoli

Dipartimento di Matematica, Università degli Studi di Salerno
Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy
Email: {mcapaldo, adicrescenzo<EMAIL_ADDRESS>ORCID:
0000-0002-0255-8935ORCID: 0000-0003-4751-7341ORCID: 0000-0002-3516-9530

(; September 19, 2023)

###### Abstract

We introduce and study the cumulative information generating function, which
provides a unifying mathematical tool suitable to deal with classical and
fractional entropies based on the cumulative distribution function and on the
survival function. Specifically, after establishing its main properties and
some bounds, we show that it is a variability measure itself that extends the
Gini mean semi-difference. We also provide (i) an extension of such a measure,
based on distortion functions, and (ii) a weighted version based on a mixture
distribution. Furthermore, we explore some connections with the reliability of
$k$-out-of-$n$ systems and with stress-strength models for multi-component
systems. Also, we address the problem of extending the cumulative information
generating function to higher dimensions.

Mathematics Subject Classification: 94A17, 60E15, 62N05

Keywords: Entropy; Proportional hazard model; Proportional reversed hazard
model; $k$-out-of-$n$ system; Stress-strength system; Variability measure.

##  1 Introduction and background

In recent years, there is a deep interest in proposing new measures of
uncertainty, in order to respond to the increasingly diversified needs of
researchers in the fields of reliability and risk analysis. At the same time,
getting lost in the vast sea of the new notions is relatively easy. To meet
these needs, in this paper we aim to propose a new generating function which
is able to recover the cumulative residual entropy and the cumulative entropy,
as well as both their generalized and fractional extensions.

If $X$ is a nonnegative absolutely continuous random variable having support
$(0,r)$, with $r\in(0,+\infty]$, and probability density function (PDF) $f$,
the _differential entropy_ of $X$ is defined as (see, for instance, Cover and
Thomas [11])

$\mathcal{H}(X)=\mathbb{E}\left[-\log f(X)\right]=-\int_{0}^{r}f(x)\log
f(x)\,\text{d}x.$ (1)

Such a measure can be obtained from the _information generating function_ ,
defined by Golomb [21] as

$\mathcal{IG}_{X}(\nu)=\mathbb{E}[(f(X))^{\nu-1}]=\int_{0}^{r}[f(x)]^{\nu}\,\text{d}x,$
(2)

where $\nu\in\mathbb{R}$ is such that the right-hand-side of (2) is finite.
Indeed, Eqs. (1) and (2) give:

$\mathcal{H}(X)=-\frac{\text{d}}{\text{d}\nu}\mathcal{IG}_{X}(\nu)\big{|}_{\nu=1}.$
(3)

Recent developments and examples of applications of information generating
functions can be found in Clark [9], Kharazmi et al. [28], and Kharazmi and
Balakrishnan [29].

We remark that the differential entropy can take negative values, whereas the
Shannon entropy of a discrete distribution is nonnegative. To avoid this
drawback, and for other reasons as mentioned in Rao et al. [41], various
alternative measures have been proposed recently. Table 1 shows some
information measures for a random variable $X$ having support $(0,r)$, with
$r\in(0,+\infty]$, having respectively cumulative distribution function (CDF)
and survival function (SF) given by

$F(x)=\mathbb{P}(X\leq x),\qquad\overline{F}(x)=1-F(x).$

We remark that the entropies presented in Table 1 can also be expressed in
terms of the cumulative hazard rate and the cumulative reversed hazard rate of
$X$, defined respectively as

$\Lambda(x)=-\log\overline{F}(x),\qquad T(x)=-\log F(x).$ (4)

These functions are involved in the _cumulative residual entropy_ introduced
in [41], and the _cumulative entropy_ (see Di Crescenzo and Longobardi [16],
[17]), given respectively in cases (i) and (ii) of Table 1. Such measures are
obtained by replacing the PDF in (1) with the SF and the CDF, respectively.
This preserves the fact that the logarithm of the probability of an event
represents the information contained in the event, in accordance with the
Shannon entropy in the discrete case.

Table 1: Information measures of interest, for a given random lifetime $X$
with support $(0,r)$ where $r\in(0,+\infty]$, with $n\in\mathbb{N}_{0}$ for
case (iii), $n\in\mathbb{N}$ for case (iv), $\nu\geq 0$ for case (v) and
$\nu>0$ for case (vi).

(i) cumulative residual entropy | (ii) cumulative entropy
---|---
$\mathcal{CRE}(X)=-\int_{0}^{r}\overline{F}(x)\log\overline{F}(x)\,\text{d}x$ | $\mathcal{CE}(X)=-\int_{0}^{r}{F}(x)\log{F}(x)\,\text{d}x$
(iii) generalized cumulative residual entropy | (iv) generalized cumulative entropy
$\mathcal{CRE}_{n}(X)=\frac{1}{n!}\int_{0}^{r}\overline{F}(x)\left[-\log\overline{F}(x)\right]^{n}\text{d}x$ | $\mathcal{CE}_{n}(X)=\frac{1}{n!}\int_{0}^{r}F(x)\left[-\log F(x)\right]^{n}\,\text{d}x$
(v) generalized fractional cumulative residual entropy | (vi) generalized fractional cumulative entropy
$\mathcal{CRE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\int_{0}^{r}\overline{F}(x)[-\log\overline{F}(x)]^{\nu}\,\text{d}x$ | $\mathcal{CE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\int_{0}^{r}F(x)[-\log F(x)]^{\nu}\,\text{d}x$

Both the cumulative residual entropy and the cumulative entropy take
nonnegative values, vanishing only in the case of degenerate random variables.
These measures are particularly suitable for describing information in
problems related to reliability theory, where $X$ denotes the random lifetime
of an item, and $x$ is the reference time. In particular, in Table 1, the
cumulative residual entropy (i) and the generalized versions (iii) and (v)
deal with events for which the uncertainty is related to the future, while the
cumulative entropies (ii), (iv) and (vi) are suitable to quantify the
information when the uncertainty is related to the past. In addition, Asadi
and Zohrevand [3] showed that $\mathcal{CRE}(X)=\mathbb{E}[\text{mrl}(X)]$,
where mrl$(X)$ is the mean residual life of $X$. Also, in [17] it is shown
that $\mathcal{CE}(X)=\mathbb{E}[\tilde{\mu}(X)]$, where $\tilde{\mu}(X)$ is
the mean inactivity time of $X$. Moreover, other applications of these
information measures can be found in Risk Theory, since the risk is strictly
related to the notion of uncertainty, see e.g. Dulac and Simon [18].

Recently, Psarrakos and Navarro [38] introduced the _generalized cumulative
residual entropy of order $n$_ of $X$, defined as in (iii) of Table 1, in
order to extend the cumulative residual entropy. A dual information measure,
known as _generalized cumulative entropy of order $n$_, was proposed by Kayal
[27] (cf. case (iv) of Table 1). Various results on these generalized measures
have been studied by Toomaj and Di Crescenzo in [46]. In particular, the
measures given in (iii) and (iv) of Table 1 play a role in the theory of point
processes. Indeed, the generalized cumulative residual entropy of order $n$,
say $\mathcal{CRE}_{n}(X)$, is equal to the mean of the $(n+1)$-th interepoch
interval of a non-homogeneous Poisson process having cumulative intensity
function given by the first of (4). Similarly, the generalized cumulative
entropy of order $n$ can be viewed as an expected spacing in lower record
values (see, for instance, Section 6 of [46]). They are also related to the
upper and lower record values densities (see, for instance, Kumar and Dangi
[34]).

Fractional versions of the above measures have been studied as well, with the
aim of disposing with more advanced mathematical tools to handle complex
systems and anomalous dynamics. Specifically, see Xiong et al. [49] and Di
Crescenzo et al. [15], for the _fractional generalized cumulative residual
entropy_ and the _fractional generalized cumulative entropy_ of $X$, given
respectively in cases (v) and (vi) of Table 1. Certain features of fractional
calculus allow these measures to better capture long-range phenomena and
nonlocal dependence in some random systems.

The entropies considered in Table 1 deal with nonnegative random variables
since they are often referred to random lifetimes of interest in reliability
theory. However they can be straightforwardly extended to the case when $X$
has a general support contained in $(l,r)$, with $-\infty\leq l<r\leq+\infty$.

The aim of this paper is to propose and study a new information generating
function that, in analogy with the functions in Eqs. (2) and (3), is able to
recover the information measures presented in Table 1. It is defined as the
integral of the product between suitable powers of the CDF and the SF.
Throughout the paper it emerges that some advantages related to the use of the
new generating function are:
– the convenience of gaining information from both the CDF and the SF of the
random variable under investigation,
– the existence of suitable applications to notions of interest in reliability
theory such as proportional hazards, odds function, order statistics, $k$-out-
of-$n$ systems, and stress-strength models for multi-component systems,
– the possibility of using it as a measure of concentration, since it is an
extension of the Gini mean semi-difference.

With reference to the latter statement, we will show also that the proposed
generating function can be extended (i) by replacing the powers of the CDF and
the SF with suitable distortion functions, and (ii) by defining a weighted
version based on a mixture distribution. Moreover, it is worth mentioning that
the proposed generating function and its generalized versions can be applied
in risk analysis. Indeed, we show that they are proper variability measures.

###  1.1 Plan of the paper

In Section 2 we define the new generating function, named cumulative
information generating function. We show that it is useful to recover the
measures given in Table 1. Moreover, we illustrate the effect of an affine
transformation of the considered random variable, and provide some connections
with the proportional hazard model, the proportional reversed hazard model,
and the odds function.

In Section 3 we use various well-known inequalities in order to obtain some
bounds for the cumulative information generating function. In addition, we
show how the cumulative information generating function is related (i) to the
Euler beta function, and (ii) to the Golomb’s information generating function
of the equilibrium random variable.

In Section 4 we discuss the connections with some notions of systems
reliability, as series and parallel systems, and $k$-out-of-$n$ systems, also
with special attention to the reliability of the multi-component stress-
strength system.

In Section 5 we introduce the above mentioned generalized information
measures, named ‘q-distorted Gini function’ and ‘weighted q-distorted Gini
function’, being related also to the Gini mean semi-difference. We also prove
that they are suitable variability measures, since in particular the
dispersive order between pairs of random variables implies the ordering
between these functions. An application to the reliability of multi-component
stress-strength systems is provided, too.

The Section 6 is concerning the extension of the cumulative information
generating function to the case of a two-dimensional random vector, with
special care to the case of independent components.

Some final remarks are then given in Section 7.

Throughout the paper, the terms increasing and decreasing are used in non-
strict sense, $\mathbb{N}$ denotes the set of positive integers, and
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. Moreover, given a distribution
function $F(x)$, we denote the right-continuous version of its inverse by
$F^{-1}(u)=\sup\\{x\colon F(x)\leq u\\}$, $u\in[0,1]$, which is also named
quantile function in statistical framework.

##  2 Cumulative information generating function

In the same spirit of Eq. (2) we now introduce a new generating function which
allows to measure the cumulative information coming both from the CDF and the
SF.

###### Definition 2.1

Let $X$ be a random variable with CDF $F(x)$ and SF $\overline{F}(x)$,
$x\in\mathbb{R}$, and let

$l=\inf\\{x\in\mathbb{R}:F(x)>0\\},\qquad
r=\sup\\{x\in\mathbb{R}:\overline{F}(x)>0\\}$ (5)

denote respectively the lower and upper limits of the support of $X$ (which
may be finite or infinite). The cumulative information generating function
(CIGF) of $X$ is defined as

$\displaystyle G_{X}:$
$\displaystyle\>D_{X}\subseteq\mathbb{R}^{2}\longrightarrow[0,+\infty)$ (6)
$\displaystyle\;\;(\alpha,\beta)\quad\;\longmapsto\;G_{X}(\alpha,\beta)=\int_{l}^{r}\left[F(x)\right]^{\alpha}\left[\overline{F}(x)\right]^{\beta}\,{\rm{d}}x$

where

$D_{X}=\\{(\alpha,\beta)\in\mathbb{R}^{2}:G_{X}(\alpha,\beta)<+\infty\\}.$

Clearly, one has $\mathbb{P}(l\leq X\leq r)=1$. Moreover, if $X$ is degenerate
then $G_{X}(\alpha,\beta)=0$, otherwise $G_{X}(\alpha,\beta)>0$.

###### Example 2.1

Let $X\sim Erlang(2,\lambda)$, with $\lambda>0$, and CDF $F(x)=1-e^{-\lambda
x}-\lambda xe^{-\lambda x}$, $x\geq 0$. From (6), recalling that $\forall
x:|x|<1$,

$(1+x)^{\alpha}=\sum_{n=0}^{+\infty}\binom{\alpha}{n}x^{n},\qquad\hbox{with
}\;\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!},$ (7)

and taking into account that

$\int_{0}^{+\infty}e^{-(n+\beta)\lambda x}(1+\lambda
x)^{n+\beta}\,\text{d}x=\frac{1}{\lambda}\,\Gamma(n+\beta+1,n+\beta)\,e^{(n+\beta)}(n+\beta)^{-(n+\beta+1)},$

with
$D_{X}=\\{(\alpha,\beta)\in\mathbb{R}^{2}:\beta\in\mathbb{R}\setminus\mathbb{Z}_{0}^{-}\\}$,
we obtain the CIGF of $X$ in series form:

$G_{X}(\alpha,\beta)=\frac{1}{\lambda}\sum_{n=0}^{+\infty}\binom{\alpha}{n}(-1)^{n}\,\Gamma(n+\beta+1,n+\beta)\,e^{n+\beta}\,(n+\beta)^{-(n+\beta+1)}.$
(8)

We remark that if $X$ is a discrete random variable with finite support
$\\{x_{1}\leq x_{2}\leq\ldots\leq x_{n}\\}$, then due to (6) the CIGF can be
expressed as a sum, i.e.

$G_{X}(\alpha,\beta)=\sum_{k=1}^{n-1}\left(P_{k}\right)^{\alpha}\left(1-P_{k}\right)^{\beta}(x_{k+1}-x_{k}),\qquad\hbox{where
\ }P_{k}=\sum_{i=1}^{k}\mathbb{P}(X=x_{i}).$

Other examples will be illustrated below.

In the next theorem we show the effect of an affine transformation. The result
follows from Definition 2.1, and recalling the relation between the CDFs of
$X$ and $Y=\gamma X+\delta$.

###### Theorem 2.1

Let $X$ be a random variable with finite CIGF. Consider the affine
transformation $Y=\gamma X+\delta$, with
$\gamma\in\mathbb{R}\setminus\\{0\\}$, $\delta\in\mathbb{R}$. Then

$G_{Y}(\alpha,\beta)=\begin{cases}\gamma\,G_{X}(\alpha,\beta),&\text{$0<\gamma<\infty$}\\\
|\gamma|\,G_{X}(\beta,\alpha),&\text{$-\infty<\gamma<0$}\end{cases}$ (9)

for $(\alpha,\beta)\in D_{X}$ if $0<\gamma<\infty$, and $(\beta,\alpha)\in
D_{X}$ if $-\infty<\gamma<0$.

###### Remark 2.1

If $X$ is absolutely continuous, with PDF $f(x)$, by setting $u=F(x)$ in the
right-hand-side of Eq. (6), the CIGF of $X$ can be expressed as

$G_{X}(\alpha,\beta)=\int_{0}^{1}u^{\alpha}(1-u)^{\beta}\frac{1}{f(F^{-1}(u))}\,{\rm
d}u,\qquad(\alpha,\beta)\in D_{X}.$ (10)

###### Remark 2.2

The CIGF of a nonnegative random variable $X$ can be regarded as a measure of
concentration. Indeed, if $X^{\prime}$ is an independent copy of $X$, we can
deduce that (see, for instance, the proof of Proposition 1 of Rao [40])

$G_{X}(1,1)=\int_{l}^{r}F(x)\overline{F}(x)\,{\rm{d}}x=\frac{1}{2}\,\mathbb{E}\left[|X-X^{\prime}|\right]=Gini(X).$
(11)

This quantity is also known as the Gini mean semi-difference, which represents
an example of coherent measure of variability with comonotonic additivity (see
Section 2.2 of Hu and Chen [26]).

Similarly as the information generating function defined in (2), the following
generating measures can be introduced as marginal versions of the CIGF.

###### Definition 2.2

Under the same assumptions of Definition 2.1, the cumulative information
generating measure and the cumulative residual information generating measure
are defined respectively by

$H_{X}(\alpha)\equiv
G_{X}(\alpha,0)=\int_{l}^{r}\left[F(x)\right]^{\alpha}\,{\rm{d}}x,\qquad\forall\;(\alpha,0)\in
D_{X}$ (12)

and

$K_{X}(\beta)\equiv
G_{X}(0,\beta)=\int_{l}^{r}\left[\overline{F}(x)\right]^{\beta}\,{\rm{d}}x,\qquad\forall\;(0,\beta)\in
D_{X}.$ (13)

We remark that when $X$ is absolutely continuous with support $(0,\infty)$,
the measure (13) has been introduced in Eq. (10) of Kharazmi and Balakrishnan
[30], denoted as $\mathcal{CIG}_{\beta}(\overline{F})$, for $\beta>0$. Under
these assumptions, other properties and the non-parametric estimation of the
function given in Eq. (13) have been studied in Smitha et al. [44].

###### Remark 2.3

If $X$ is a random variable with finite CIGF and symmetric CDF, in the sense
that for some $m\in\mathbb{R}$ one has $F(m+x)=\overline{F}(m-x)$
$\forall\,x\in\mathbb{R}$, then
(i) $G_{X}(\alpha,\beta)=G_{X}(\beta,\alpha)$ for all $(\alpha,\beta)\in
D_{X}$;
(ii) from Eqs. (12) and (13) we have $H_{X}(\alpha)=K_{X}(\alpha)$ for all
$(\alpha,0)\in D_{X}$;
(iii) under the assumptions of Theorem 2.1, Eq. (9) becomes
$G_{Y}(\alpha,\beta)=|\gamma|\,G_{X}(\alpha,\beta)$, $\gamma\in\mathbb{R}$.

Recalling the measures (i) and (ii) of Table 1, now we can show that the
cumulative residual entropy and the cumulative entropy can be obtained from
the CIGF.

###### Proposition 2.1

Let $X$ be a random variable having finite CIGF $G_{X}(\alpha,\beta)$, for
$(\alpha,\beta)\in D_{X}$. If $(0,1)\in D_{X}$, then

$\mathcal{CRE}(X)=-\frac{\partial}{\partial\beta}G_{X}(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1}.$

If $(1,0)\in D_{X}$, then

$\mathcal{CE}(X)=-\frac{\partial}{\partial\alpha}G_{X}(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0}.$

  * Proof.

The stated results follow from Eq. (6), by differentiation under the integral
sign.

$\Box$

Let us now obtain a similar relation for the generalized cumulative residual
entropy and the generalized cumulative entropy (cf. cases (iii) and (iv) of
Table 1).

###### Proposition 2.2

Let $X$ be a random variable having finite CIGF $G_{X}(\alpha,\beta)$, for
$(\alpha,\beta)\in D_{X}$. If $(0,1)\in D_{X}$, then

$\mathcal{CRE}_{n}(X)=\frac{(-1)^{n}}{n!}\frac{\partial^{n}}{\partial\beta^{n}}G_{X}(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1},\qquad
n\in\mathbb{N}_{0}.$ (14)

If $(1,0)\in D_{X}$, then

$\mathcal{CE}_{n}(X)=\frac{(-1)^{n}}{n!}\frac{\partial^{n}}{\partial\alpha^{n}}G_{X}(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0},\qquad
n\in\mathbb{N}.$ (15)

  * Proof.

The proof of (15) is analogous to Proposition 2.1, by using this identity:

$\frac{\partial^{n}}{\partial\alpha^{n}}G_{X}(\alpha,\beta)=\int_{l}^{r}\left(\log
F(x)\right)^{n}\left[F(x)\right]^{\alpha}\left[\overline{F}(x)\right]^{\beta}\,\text{d}x.$
(16)

In the same way we obtain Eq. (14).

$\Box$

In order to extend the above relations to the case of the generalized
fractional cumulative residual entropy and the generalized fractional
cumulative entropy, given respectively in cases (v) and (vi) of Table 1, let
us now recall briefly the expression of the Caputo fractional derivatives
(see, for instance, Kilbas et al. [31]). Specifically, given a function
$y(x_{1},x_{2})$, we consider

$\left(\prescript{C}{}{D_{-,x_{1}}^{\nu}y}\right)(x_{1},x_{2})=\frac{(-1)^{n}}{\Gamma(n-\nu)}\int_{x_{1}}^{+\infty}\frac{y^{(n)}(t,x_{2})}{(t-x_{1})^{\nu+1-n}}\,\text{d}t,\qquad
x_{1},x_{2}\in\mathbb{R},$

that is the _left-sided Caputo partial fractional derivative with respect to
$x_{1}$ of order $\nu$_ on the whole axis $\mathbb{R}$, where
$\nu\in\mathbb{C}$ with $\text{Re}(\nu)>0$, $\nu\notin\mathbb{N}$ and
$n=\lfloor\text{Re}(\nu)\rfloor+1$.

###### Proposition 2.3

Let $X$ be a random variable having finite CIGF $G_{X}(\alpha,\beta)$, for
$(\alpha,\beta)\in D_{X}$. If $(0,1)\in D_{X}$, then

$\mathcal{CRE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-,\beta}^{\nu}G_{X}}\right)(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1},\qquad\nu>0.$
(17)

If $(1,0)\in D_{X}$, then

$\mathcal{CE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-,\alpha}^{\nu}G_{X}}\right)(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0},\qquad\nu>0.$
(18)

  * Proof.

We show only the proof of Eq. (18) because Eq. (17) can be derived similarly.
From (16) we obtain

$\displaystyle\left(\prescript{C}{}{D_{-,\alpha}^{\nu}G_{X}}\right)(\alpha,\beta)=$
$\displaystyle=\frac{(-1)^{n}}{\Gamma(n-\nu)}\int_{\alpha}^{+\infty}\frac{1}{(t-\alpha)^{\nu+1-n}}\frac{\partial^{n}}{\partial
t^{n}}G_{X}(t,\beta)\,\text{d}t$
$\displaystyle=\frac{(-1)^{n}}{\Gamma(n-\nu)}\int_{\alpha}^{+\infty}\frac{1}{(t-\alpha)^{\nu+1-n}}\left(\int_{l}^{r}\left(\log
F(x)\right)^{n}\left[F(x)\right]^{t}\left[\overline{F}(x)\right]^{\beta}\,\text{d}x\right)\,\text{d}t$
$\displaystyle=\frac{(-1)^{n}}{\Gamma(n-\nu)}\int_{l}^{r}\left(\log
F(x)\right)^{n}\left[\overline{F}(x)\right]^{\beta}\left(\int_{\alpha}^{+\infty}\frac{\left[F(x)\right]^{t}}{(t-\alpha)^{\nu+1-n}}\,\text{d}t\right)\,\text{d}x,$

where the last equality is obtained by use of Fubini’s theorem. By placing
$t-\alpha=z$ and $\gamma=-z\log F(x)$, we have

$\displaystyle\int_{\alpha}^{+\infty}\frac{\left[F(x)\right]^{t}}{(t-\alpha)^{\nu+1-n}}\,\text{d}t$
$\displaystyle=\left[F(x)\right]^{\alpha}\int_{0}^{+\infty}\left[F(x)\right]^{z}z^{n-\nu-1}\,\text{d}z$
$\displaystyle=\left[F(x)\right]^{\alpha}\int_{0}^{+\infty}e^{-\gamma}\gamma^{n-\nu-1}\left(-\log
F(x)\right)^{-n+\nu}\,\text{d}\gamma$
$\displaystyle=\left[F(x)\right]^{\alpha}\Gamma(n-\nu)\left(-\log
F(x)\right)^{-n+\nu}.$

Finally we deduce

$\displaystyle\left(\prescript{C}{}{D_{-,\alpha}^{\nu}G_{X}}\right)(\alpha,\beta)=\int_{l}^{r}\left(-\log
F(x)\right)^{\nu}\left[\overline{F}(x)\right]^{\beta}\left[F(x)\right]^{\alpha}\,\text{d}x,$

so that Eq. (18) follows by taking $\alpha=1$ and $\beta=0$.

$\Box$

###### Remark 2.4

Recently, Kharazmi and Balakrishnan [30] noted that
$\mathcal{CRE}(X)=-\frac{\text{d}}{\text{d}\beta}K_{X}(\beta)\big{|}_{\beta=1}$.
Similarly, we obtain $\mathcal{CE}(X)=-\frac{{\rm d}}{{\rm
d}\alpha}H_{X}(\alpha)\big{|}_{\alpha=1}$. Moreover, for the generalized
versions and for the fractional versions we have respectively

$\mathcal{CRE}_{n}(X)=\frac{(-1)^{n}}{n!}\frac{{\rm d}^{n}}{{\rm
d}\beta^{n}}K_{X}(\beta)\Big{|}_{\beta=1},\qquad\mathcal{CE}_{n}(X)=\frac{(-1)^{n}}{n!}\frac{{\rm
d}^{n}}{{\rm d}\alpha^{n}}H_{X}(\alpha)\Big{|}_{\alpha=1},$
$\mathcal{CRE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-}^{\nu}K_{X}}\right)(\beta)\Big{|}_{\beta=1},\qquad\mathcal{CE}_{\nu}(X)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-}^{\nu}H_{X}}\right)(\alpha)\Big{|}_{\alpha=1}.$

Now we recall two important models that are largely adopted in survival
analysis and reliability theory. Let $X$ be a random lifetime with CDF $F(x)$
and SF $\overline{F}(x)$. The proportional hazard model (see for instance Cox
[10], Kumar and Klefsjö [35]) is expressed by a random lifetime
$X^{*}_{\gamma}$ with SF

$\overline{F}_{X^{*}_{\gamma}}(x)=\left[\overline{F}(x)\right]^{\gamma},\qquad\gamma\in\mathbb{R}^{+}.$
(19)

Similarly, the proportional reversed hazard model (see for instance Di
Crescenzo [14], Gupta and Gupta [24], Gupta et al. [25]) is expressed by a
random lifetime $\hat{X}_{\theta}$ with CDF

$F_{\hat{X}_{\theta}}(x)=\left[F(x)\right]^{\theta},\qquad\theta\in\mathbb{R}^{+}.$
(20)

Recently, modified versions of these models have been studied by Das and Kayal
[12].

###### Remark 2.5

The measures given in Definition 2.2 satisfy the following relations.

(i) Under the proportional hazard model (19) we have

$K_{X^{*}_{\gamma}}(\beta)=K_{X}(\gamma\beta)\qquad\forall\,\gamma\in\mathbb{R}^{+}\;\;\hbox{s.t.}\;\;(0,\gamma\beta)\in
D_{X}.$

(ii) Under the proportional reversed hazard model (20) we have

$H_{\hat{X}_{\theta}}(\alpha)=H_{X}(\theta\alpha)\qquad\forall\,\theta\in\mathbb{R}^{+}\;\;\hbox{s.t.}\;\;(\theta\alpha,0)\in
D_{X}.$

###### Remark 2.6

If $X$ is a random lifetime such that $D_{X}\subseteq(\mathbb{R}^{+})^{2}$,
then recalling the Definition 2.1 and Eqs. (19) and (20), the CIGF of $X$ can
be expressed as

$G_{X}(\alpha,\beta)=\int_{l}^{r}F_{\hat{X}_{\alpha}}(x)\,\overline{F}_{X^{*}_{\beta}}(x)\,\text{d}x.$

We now recall another useful concept. Let $X$ be a random variable with CDF
and SF denoted by $F(x)$ and $\overline{F}(x)$, respectively. For all
$x\in(l,r)$ the _odds function_ of $X$ is (cf. Kirmani and Gupta [32])

$\theta(x)=\frac{\overline{F}(x)}{F(x)}.$ (21)

This function represents the ratio of the probability of an event occurring to
the probability of its not occurring, and always assumes nonnegative finite
values. It is used in reliability theory, because it quantifies the strength
of the association between the failure of a system after time $x$ and before
time $x$. Due to Eq. (21), we can express the CIGF of $X$ in terms of the odds
function in two equivalent useful ways:

$G_{X}(\alpha,\beta)=\int_{l}^{r}[F(x)]^{\alpha+\beta}[\theta(x)]^{\beta}\,\text{d}x=\int_{l}^{r}\left[\theta(x)\right]^{-\alpha}\left[\overline{F}(x)\right]^{\alpha+\beta}\,\text{d}x.$

Hence, when the parameters $\alpha$ and $\beta$ may take negative values such
that $\alpha+\beta=0$, then the CIGF of $X$ can be expressed in terms of the
odds function as

$G_{X}(-\beta,\beta)=\int_{l}^{r}[{\theta(x)}]^{\beta}\,\text{d}x\qquad\forall\,\beta\in
D_{X,\theta}$ (22)

where

$D_{X,\theta}=\\{\beta\in\mathbb{R}\colon G_{X}(-\beta,\beta)<+\infty\\}.$

Table 2 shows various examples of the CIGF expressed in terms of the Euler
Beta function $B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\,\text{d}t$ or the
incomplete Beta function
$B\left(p;x,y\right)=\int_{0}^{p}t^{x-1}(1-t)^{y-1}\,\text{d}t$, $p\in[0,1]$.

Finally, by recalling Eq. (11), we remark that for $\alpha=\beta=1$ Eq. (8)
and the examples in Table 2 are in agreement with Giorgi and Nadarajah [19].

Table 2: The CIGF for some notable distributions.

Distribution | $D_{X}$ | $D_{X,\theta}$ | $G_{X}(\alpha,\beta)$
---|---|---|---
$X\sim\text{Bernoulli}(p)$ | $\mathbb{R}^{2}$ | $\mathbb{R}$ | $(1-p)^{\alpha}\,p^{\beta}=\frac{{\rm d}}{{\rm d}p}B\left(p;\beta+1,\alpha+1\right)$
$X\sim\text{Unif}(l,r)$ | $\\{(\alpha,\beta)\in\mathbb{R}^{2}:-1<\alpha<\infty,-1<\beta<\infty\\}$ | $(-1,1)$ | $(r-l)B(\alpha+1,\beta+1)$
$X\sim\text{Power}(\theta)$ | $\\{(\alpha,\beta)\in\mathbb{R}^{2}:-\frac{1}{\theta}<\alpha<\infty,-1<\beta<\infty\\}$ | $\left(-\frac{1}{\theta},\frac{1}{\theta}\right)$ | $\frac{1}{\theta}\;B\left(\alpha+\frac{1}{\theta},\beta+1\right)$
$X\sim\text{Exp}(\lambda)$ | $\\{(\alpha,\beta)\in\mathbb{R}^{2}:-1<\alpha<\infty,0<\beta<\infty\\}$ | $(0,1)$ | $\frac{1}{\lambda}B(\alpha+1,\beta)$
$X\sim\text{Laplace}(0,\lambda)$ | $\\{(\alpha,\beta)\in\mathbb{R}^{2}:0<\alpha<\infty,\;0<\beta<\infty\\}$ | $\emptyset$ | $\frac{\lambda}{2}\left[B\left(\frac{1}{2};\alpha,\beta+1\right)+B\left(\frac{1}{2};\alpha+1,\beta\right)\right]$

##  3 Inequalities and further results

In this section, we obtain some bounds and further results regarding the CIGF.
Specifically, we first refer to well-known inequalities named after Chernoff,
Bernoulli, Minkowski and Hölder’s (see, for istance Schilling [42]).

Hereafter, thanks to the Chernoff’s inequalities we recover some bounds for
the CIGF in terms of the moment generating function (MGF) of $X$, denoted by
$M_{X}(s)=\mathbb{E}(e^{sX})$, $s\in\mathbb{R}$.

###### Proposition 3.1

Let $X$ be a nonnegative random variable with support $(0,r)$, where
$r\in(0,+\infty)$, and having finite CIGF $G_{X}(\alpha,\beta)$, for
$(\alpha,\beta)\in D_{X}$. Assume that the MGF of $X$ satisfies
$M_{X}(s)<+\infty$ for all $s\in(-s_{0},s_{0})$, with $s_{0}>0$. Then,
(i) for all $s_{1}\in(-s_{0},0)$, $s_{2}\in(0,s_{0})$ and $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{+})^{2}$, one has

$G_{X}(\alpha,\beta)\leq g(r;\alpha,\beta,{\bf
s})\left[M_{X}(s_{1})\right]^{\alpha}\left[M_{X}(s_{2})\right]^{\beta},$ (23)

(ii) for all $s_{1}\in(-s_{0},0)$, $s_{2}\in(0,s_{0})$ and $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{-})^{2}$, one has

$G_{X}(\alpha,\beta)\geq g(r;\alpha,\beta,{\bf
s})\left[M_{X}(s_{1})\right]^{\alpha}\left[M_{X}(s_{2})\right]^{\beta},$ (24)

where

$g(r;\alpha,\beta,{\bf
s})=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{\alpha s_{1}+\beta
s_{2}}\left[1-e^{-(\alpha s_{1}+\beta s_{2})r}\right],&\hbox{ if }\alpha
s_{1}+\beta s_{2}\neq 0,\\\\[5.69054pt] r,&\hbox{ if }\alpha s_{1}+\beta
s_{2}=0.\end{array}\right.$

  * Proof.

Applying Chernoff’s inequalities $\mathbb{P}(X\leq x)\leq
e^{-s_{1}x}M_{X}(s_{1})$ and $\mathbb{P}(X\geq x)\leq
e^{-s_{2}x}M_{X}(s_{2})$, for $x>0$, to Eq. (6), by the effect of integration,
for all $s_{1}\in(-s_{0},0)$, $s_{2}\in(0,s_{0})$ and $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{+})^{2}$, it follows that

$G_{X}(\alpha,\beta)\leq\int_{0}^{r}\left[e^{-s_{1}x}M_{X}(s_{1})\right]^{\alpha}\left[e^{-s_{2}x}M_{X}(s_{2})\right]^{\beta}\,\text{d}x.$

Few calculations give Eq. (23). For $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{-})^{2}$, Eq. (24) can be obtained similarly.

$\Box$

The case when $X$ has support $(0,+\infty)$ can be easily derived as follows.

###### Corollary 3.1

Let $X$ be a nonnegative random variable with support $(0,+\infty)$, and
having finite CIGF $G_{X}(\alpha,\beta)$, for $(\alpha,\beta)\in D_{X}$.
Assume that the MGF of $X$ satisfies $M_{X}(s)<+\infty$ for all
$s\in(-s_{0},s_{0})$, with $s_{0}>0$. Then,
(i) for all $s_{1}\in(-s_{0},0)$, $s_{2}\in(0,s_{0})$ and $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{+})^{2}$ such that $\alpha s_{1}+\beta s_{2}>0$, one has

$G_{X}(\alpha,\beta)\leq\frac{1}{\alpha s_{1}+\beta
s_{2}}\left[M_{X}(s_{1})\right]^{\alpha}\left[M_{X}(s_{2})\right]^{\beta},$
(25)

(ii) for all $s_{1}\in(-s_{0},0)$, $s_{2}\in(0,s_{0})$ and $(\alpha,\beta)\in
D_{X}\cap(\mathbb{R}^{-})^{2}$ such that $\alpha s_{1}+\beta s_{2}>0$, one has

$G_{X}(\alpha,\beta)\geq\frac{1}{\alpha s_{1}+\beta
s_{2}}\left[M_{X}(s_{1})\right]^{\alpha}\left[M_{X}(s_{2})\right]^{\beta}.$

The following example provides an application of the previous results.

###### Example 3.1

Let $X\sim Erlang(2,\lambda)$, $\lambda>0$, as in Example 2.1, with MGF
$M_{X}(s)=\lambda^{2}(\lambda-s)^{-2}$, for $s<\lambda$. Thanks to Eqs. (8)
and (25), some calculations give, for $(\alpha,\beta)\in(\mathbb{R}^{+})^{2}$,

$G_{X}(\alpha,\beta)\leq\inf_{\begin{subarray}{c}(s_{1},s_{2})\in(-\lambda,0)\times(0,\lambda)\\\
\alpha s_{1}+\beta s_{2}>0\end{subarray}}\,\frac{1}{\alpha s_{1}+\beta
s_{2}}\cdot\frac{\lambda^{2(\alpha+\beta)}}{(\lambda-s_{1})^{2\alpha}(\lambda-
s_{2})^{2\beta}}=\frac{1}{\lambda}\frac{1}{2^{2\beta}}\left(\frac{1+2\beta}{\beta}\right)^{1+2\beta}.$

Let us now express some upper bounds for the CIGF in terms of the measures
introduced in Definition 2.2.

###### Proposition 3.2

Under the assumptions specified in Definition 2.1, the CIGF of a random
variable $X$ satisfies the following inequalities:

$G_{X}(\alpha,\beta)\leq K_{X}(\beta)-\alpha
K_{X}(\beta+1)\qquad\forall\,(\alpha,\beta)\in
D_{X}\cap[0,1]\times\mathbb{R},$ $G_{X}(\alpha,\beta)\leq H_{X}(\alpha)-\beta
H_{X}(\alpha+1)\qquad\forall\,(\alpha,\beta)\in
D_{X}\cap\mathbb{R}\times[0,1].$

  * Proof.

Due to Bernoulli’s inequality with real exponents, for all $x\in\mathbb{R}$ it
follows that

$\left[1-\overline{F}(x)\right]^{\alpha}\leq
1-\alpha\overline{F}(x)\qquad\forall\,\alpha\in[0,1]$

and

$\left[1-F(x)\right]^{\beta}\leq 1-\beta F(x)\qquad\forall\,\beta\in[0,1].$

Hence, the thesis immediately follows from Definitions 2.1 and 2.2.

$\Box$

Hereafter we use the Minkowski’s inequality to obtain suitable bounds for the
measures introduced in Definition 2.2 and for $G_{X}(\gamma,\gamma)$.

###### Proposition 3.3

Under the assumptions specified in Definition 2.1 and Definition 2.2, if $X$
has finite support in $(l,r)$,
(i) for all $\gamma\geq 1$ such that $(\gamma,0)\in D_{X}$ and $(0,\gamma)\in
D_{X}$ we have

$\left[\left(r-l\right)^{\frac{1}{\gamma}}-\left[H_{X}(\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma}\leq
K_{X}(\gamma)\leq\left[\left(r-l\right)^{\frac{1}{\gamma}}+\left[H_{X}(\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma},$
$\left[\left(r-l\right)^{\frac{1}{\gamma}}-\left[K_{X}(\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma}\leq
H_{X}(\gamma)\leq\left[\left(r-l\right)^{\frac{1}{\gamma}}+\left[K_{X}(\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma};$

(ii) for all $\gamma\geq 1$ such that $(0,\gamma)\in D_{X}$ we have

$G_{X}(\gamma,\gamma)\leq\left[\left[K_{X}(\gamma)\right]^{\frac{1}{\gamma}}+\left[K_{X}(2\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma};$

(iii) for all $\gamma\geq 1$ such that $(\gamma,0)\in D_{X}$ we have

$G_{X}(\gamma,\gamma)\leq\left[\left[H_{X}(\gamma)\right]^{\frac{1}{\gamma}}+\left[H_{X}(2\gamma)\right]^{\frac{1}{\gamma}}\right]^{\gamma}.$

  * Proof.

By applying Minkowski’s inequality, for $\gamma\geq 1$ we have

$\displaystyle\left[K_{X}(\gamma)\right]^{\frac{1}{\gamma}}=\left[\int_{l}^{r}\left[1-F(x)\right]^{\gamma}\,\text{d}x\right]^{\frac{1}{\gamma}}$
$\displaystyle\leq\left(r-l\right)^{\frac{1}{\gamma}}+\left[\int_{l}^{r}\left[F(x)\right]^{\gamma}\,\text{d}x\right]^{\frac{1}{\gamma}}$
$\displaystyle=\left(r-l\right)^{\frac{1}{\gamma}}+\left[H_{X}(\gamma)\right]^{\frac{1}{\gamma}},$

and also

$\displaystyle\left(r-l\right)^{\frac{1}{\gamma}}=\left[\int_{l}^{r}\left[F(x)+\overline{F}(x)\right]^{\gamma}\,\text{d}x\right]^{\frac{1}{\gamma}}$
$\displaystyle\leq\left[\int_{l}^{r}\left[F(x)\right]^{\gamma}\,\text{d}x\right]^{\frac{1}{\gamma}}+\left[\int_{l}^{r}\left[\overline{F}(x)\right]^{\gamma}\,\text{d}x\right]^{\frac{1}{\gamma}}$
$\displaystyle=\left[H_{X}(\gamma)\right]^{\frac{1}{\gamma}}+\left[K_{X}(\gamma)\right]^{\frac{1}{\gamma}}.$

Combining the two latter inequalities we obtain the bounds for
$K_{X}(\gamma)$. The other relations can be obtained in the same way, by
taking into account that, for $\gamma\geq 1$, if $(0,\gamma)\in D_{X}$ then
$(0,2\gamma)\in D_{X}$, and if $(\gamma,0)\in D_{X}$ then $(2\gamma,0)\in
D_{X}$.

$\Box$

We now prove an upper bound for $G_{X}(\alpha,\beta)$, for $\alpha+\beta=1$,
making use of the Hölder’s inequality.

###### Proposition 3.4

Under the assumptions specified in Definition 2.1, let $X$ have finite support
in $(l,r)$, with $(\theta,1-\theta)\in D_{X}$ for all $\theta\in(0,1)$. Then,

$G_{X}(\theta,1-\theta)\leq\left[r-\mathbb{E}(X)\right]^{\theta}\left[\mathbb{E}(X)-l\right]^{1-\theta}\qquad\forall\;\theta\in(0,1).$
(26)

  * Proof.

Due to the Hölder’s inequality with conjugate exponents
$\frac{1}{\theta},\frac{1}{1-\theta}$, for all $\theta\in(0,1)$ we have

$G_{X}(\theta,1-\theta)\leq\left(\int_{l}^{r}F(x)\,\text{d}x\right)^{\theta}\left(\int_{l}^{r}\overline{F}(x)\,\text{d}x\right)^{1-\theta},$

this yielding Eq. (26).

$\Box$

We remark that Eq. (26) is satisfied as equality when $F(x)=\overline{F}(x)$
$\forall\,x\in(l,r)$, i.e. when $\mathbb{P}(X=l)=\mathbb{P}(X=r)=1/2$.
Moreover, we note that the right-hand-side of Eq. (26) can be rewritten by
taking into account that, under the given assumptions,

$r-\mathbb{E}(X)=H_{X}(1),\qquad\mathbb{E}(X)-l=K_{X}(1).$

Hereafter we show that the CIGF of an absolutely continuous random variable
can be expressed as the product of the Euler beta function and the expected
value of a suitably transformed beta-distributed random variable.

###### Proposition 3.5

If $X$ is an absolutely continuous random variable with PDF $f$, CDF $F$ and
finite CIGF, then,

$G_{X}(\alpha,\beta)=B(\alpha+1,\beta+1)\,\mathbb{E}\left[r(Y)\right],$

where $Y\sim{\rm Beta}(\alpha+1,\beta+1)$ is independent from $X$, with
$r(Y)=[f(F^{-1}(Y))]^{-1}$.

  * Proof.

The thesis is obtained making use of Eq. (10) and recalling the PDF of
$Y\sim{\rm Beta}(\alpha+1,\beta+1)$.

$\Box$

The above result collects various features of the CIGF, i.e. the relations (i)
to the Beta distribution, which reflects the form of the right-hand-side of
(6), and (ii) to the transformation $f(F^{-1}(\cdot))$, which plays a relevant
role in the context of variability measures as developed in Section 5.

We conclude this section by relating the CIGF to a series involving the
Golomb’s information generating function of the equilibrium random variable.
To this aim, we recall that for a nonnegative random variable $X$, with SF
$\overline{F}(x)$ and expected value $\mathbb{E}[X]\in(0,+\infty)$, the
equilibrium random variable of $X$ is a nonnegative absolutely continuous
random variable, denoted as $X_{e}$, whose PDF is given by

$f_{e}(x)=\frac{\overline{F}(x)}{\mathbb{E}[X]},\qquad x>0.$ (27)

Recalling Eq. (2), hereafter we denote by $\mathcal{IG}_{X_{e}}$ the Golomb’s
information generating function of the equilibrium random variable $X_{e}$.

###### Proposition 3.6

If $X$ is a nonnegative random variable having expected value
$\mathbb{E}[X]\in(0,+\infty)$ and with finite CIGF, then

$G_{X}(\alpha,\beta)=\sum_{n=0}^{\infty}\binom{\alpha}{n}(-1)^{n}(\mathbb{E}[X])^{n+\beta}\,\mathcal{IG}_{X_{e}}(n+\beta)\qquad\forall\;(\alpha,\beta)\in
D_{X},$

where $X_{e}$ is the equilibrium random variable of $X$.

  * Proof.

Denoting by $(0,r)$ the support of $X$, with $r\in(0,+\infty]$, from Eqs. (6)
and (27) it follows that, for all $(\alpha,\beta)\in D_{X}$

$G_{X}(\alpha,\beta)=\int_{0}^{r}\left(1-f_{e}(x)\,\mathbb{E}[X]\right)^{\alpha}(f_{e}(x)\,\mathbb{E}[X])^{\beta}\;\text{d}x.$
(28)

Since $\left|f_{e}(x)\,\mathbb{E}[X]\right|<1$, due to Eq. (7) we have

$(1-f_{e}(x)\,\mathbb{E}[X])^{\alpha}=\sum_{n=0}^{\infty}\binom{\alpha}{n}(-1)^{n}\left(f_{e}(x)\,\mathbb{E}[X]\right)^{n}.$

By replacing the latter equation in (28), the thesis immediately follows from
Eq. (2).

$\Box$

##  4 Connections with systems reliability

In this section we relate some results exploited above to notions of interest
in reliability theory.

Several applied problems involve complex systems consisting of many
components. Here we focus on systems formed by $n$ components, where
$X_{1},X_{2},\dots,X_{n}$ describe the random lifetimes of each component. We
assume that they are independent and identically distributed (i.i.d.), with
common CDF $F(x)$ and SF $\overline{F}(x)$. As well known, a parallel system
continues to work until the last component fails, and thus its lifetime is
described by the sample maximum

$X_{(n:n)}=\max\\{X_{1},X_{2},\dots,X_{n}\\},$

which has CDF

$F_{(n:n)}(x)=\mathbb{P}(X_{(n:n)}\leq x)=\left(F(x)\right)^{n},\qquad
x\in\mathbb{R}.$ (29)

Similarly, a series system fails as soon as the first component stops working,
and thus its lifetime is described by the sample minimum

$X_{(1:n)}=\min\\{X_{1},X_{2},\dots,X_{n}\\},$

that possesses SF

$\overline{F}_{(1:n)}(x)=\mathbb{P}(X_{(1:n)}>x)=\left(\overline{F}(x)\right)^{n},\qquad
x\in\mathbb{R}.$ (30)

###### Remark 4.1

Let $n\in\mathbb{N}$. Recalling Definition 2.2, from Eqs. (29) and (30) it
immediately follows that

$H_{X_{(n:n)}}(\alpha)=H_{X}(n\alpha),\qquad\forall\;(n\alpha,0)\in D_{X},$
$K_{X_{(1:n)}}(\beta)=K_{X}(n\beta),\qquad\forall\;(0,n\beta)\in D_{X},$

where $H_{X}$ and $K_{X}$ denote respectively the cumulative information
generating measure and the cumulative residual information generating measure
of $X_{i}$.

We now focus on the expression of the CIGF for order statistics $X_{(n:n)}$
and $X_{(1:n)}$.

###### Proposition 4.1

For $n\in\mathbb{N}$, let $X_{1},X_{2},\dots,X_{n}$ be a random sample formed
by i.i.d. random lifetimes having finite cumulative information generating
measure $H_{X}$ and cumulative residual information generating measure
$K_{X}$. Then, the CIGF of the order statistics $X_{(n:n)}$ and $X_{(1:n)}$
can be expressed respectively as

$G_{X_{(n:n)}}(\alpha,\beta)=\sum_{i=0}^{\infty}(-1)^{i}\binom{\beta}{i}H_{X}\left(n(i+\alpha)\right),\qquad\forall\;(\alpha,\beta)\in
D_{X_{(n:n)}}$ (31)

and

$G_{X_{(1:n)}}(\alpha,\beta)=\sum_{j=0}^{\infty}(-1)^{j}\binom{\alpha}{j}K_{X}\left(n(j+\beta)\right),\qquad\forall\;(\alpha,\beta)\in
D_{X_{(1:n)}}.$ (32)

  * Proof.

For simplicity, assume that the support of $X$ is $(0,r)$. Recalling Eq. (6),
from Eqs. (29) and (7) we have

$G_{X_{(n:n)}}(\alpha,\beta)=\int_{0}^{r}\left[F(x)\right]^{n\alpha}\left[1-[{F}(x)]^{n}\right]^{\beta}\,\text{d}x=\sum_{i=0}^{\infty}(-1)^{i}\binom{\beta}{i}\int_{0}^{r}\left[{F}(x)\right]^{n(i+\alpha)}\,\text{d}x.$

The right-hand-side of (31) then follows making use of (12). Eq. (32) can be
obtained similarly.

$\Box$

It is well known that a system with $n$ independent components is said to be a
_$k$ -out-of-$n$ system_ when it works if and only if at least $k$ components
work (see, for instance, Boland and Proschan [7]). Clearly, if $k=1$ we have a
parallel system, while for $k=n$ we have a series system. For any $k$, the
lifetime of the $k$-out-of-$n$ system formed by components with i.i.d.
lifetimes is expressed as the corresponding $k$-th order statistic. This
allows to express the reliability and the information content of this kind of
systems in a tractable way.

###### Remark 4.2

Consider a $k$-out-of-$n$ system formed by $n$ components with i.i.d. random
lifetimes, for $n\in\mathbb{N}$. Denoting by $X_{(k:n)}$ the corresponding
$k$-th order statistic, which in turn gives the system lifetime, the CIGF of
$X_{(k:n)}$ can be expressed in terms of the cumulative information generating
measures. Indeed, similarly as Proposition 4.1, recalling Eqs. (12) and (13)
one has the following two equivalent expressions

$G_{X_{(k:n)}}(\alpha,\beta)=\sum_{i=0}^{\infty}(-1)^{i}\binom{\beta}{i}H_{X_{(k:n)}}(i+\alpha),$
$G_{X_{(k:n)}}(\alpha,\beta)=\sum_{j=0}^{\infty}(-1)^{j}\binom{\alpha}{j}K_{X_{(k:n)}}(j+\beta),$

for all $(\alpha,\beta)\in D_{X_{(k:n)}}$. Moreover, the mean of $X_{(k:n)}$
can be expressed in terms of the CIGF of $X$ as

$\mathbb{E}\left[X_{(k:n)}\right]=\sum_{j=0}^{k-1}\binom{n}{j}\int_{l}^{r}\left[F(x)\right]^{j}\left[\overline{F}(x)\right]^{n-j}\,{\rm{d}}x=\sum_{j=0}^{k-1}\binom{n}{j}\,G_{X}(j,n-j).$

Clearly, for $k=1$ we have
$\mathbb{E}\left[X_{(1:n)}\right]=G_{X}(0,n)=K_{X}(n)$.

###  4.1 Stress-strength models for multi-component systems

A further connection of the CIGF with systems reliability arises in the
analysis of stress-strength models for multi-component systems. Let us
consider a system with $n$ components having i.i.d. strengths
$X_{1},X_{2},\ldots,X_{n}$ having common CDF $F(x)$. Assume that each
component is stressed according to an independent random stress $T$ having CDF
$F_{T}(x)$. Moreover, suppose that the system survives if and only if the
components strengths are greater than the stress by at least $k$ out of $n$
($1\leq k\leq n$). Then, the reliability of the considered multi-component
stress-strength system is given by (cf. Bhattacharyya and Johnson [5])

$\displaystyle R_{k,n}$ $\displaystyle=$ $\displaystyle\mathbb{P}[\hbox{at
least $k$ of the $(X_{1},X_{2},\ldots,X_{n})$ exceed $T$}]$ (33)
$\displaystyle=$ $\displaystyle\sum_{j=k}^{n}{n\choose
j}\int_{-\infty}^{+\infty}[1-F(t)]^{j}[F(t)]^{n-j}\,{\rm d}F_{T}(t),$

with $R_{0,n}=1$. See also, for instance, the recent contribution by Kohansal
and Shoaee [33] on the statistical inference of multicomponent stress-strength
reliability under suitable censored samples.

For instance, if $T$ is distributed as $X_{1}$ then it is not hard to see that

$R_{k,n}=1-\frac{k}{n+1},\qquad 0\leq k\leq n.$ (34)

The following result is a straightforward consequence of Eqs. (6) and (33).

###### Proposition 4.2

Let $X_{1},X_{2},\ldots,X_{n},T$ have common finite support $(l,r)$, with
$X_{1},X_{2},\ldots,X_{n}$ i.i.d. If $T$ is uniformly distributed over
$(l,r)$, then

$R_{k,n}=\frac{1}{r-l}\sum_{j=k}^{n}{n\choose j}G_{X}(n-j,j),\qquad 0\leq
k\leq n.$

An iterative formula allows us to evaluate the reliability of the multi-
component stress-strength system as follows, under the assumptions of
Proposition 4.2:

$R_{k+1,n}=R_{k,n}-\frac{1}{r-l}{n\choose k}G_{X}(n-k,k),\qquad 0\leq k\leq
n-1.$

As example, if $X$ has Power$(\theta)$ distribution then from Proposition 4.2
and Table 2 after few calculations one has

$R_{k,n}=\frac{\Gamma(n+1)\,\Gamma(n-k+1+\frac{1}{\theta})}{\Gamma(n+1+\frac{1}{\theta})\,\Gamma(n-k+1)},\qquad
0\leq k\leq n,\quad\theta>0.$ (35)

Note that the expression in (35) can be also represented as a ratio of
Pochhammer symbols, or as an infinite product (cf. Eq. 8.325.1 of Gradshteyn
and Ryzhik [22]). Clearly, if $\theta=1$ then Eq. (35) reduces to Eq. (34).
Figure 1 shows some plots of $R_{k,n}$ as given in (35).

Figure 1: Plots of the reliability of the multi-component stress-strength
system with underlying Power$(\theta)$ distribution as in Eq. (35) for $0\leq
k\leq n$, with $n=50$ (left) and $n=100$ (right), with $\theta=0.1,0.5,1,2,10$
(from bottom to top).

##  5 Generalized Gini functions

This section is devoted to the analysis of a generalized version of the CIGF.
Specifically, we aim to extend the Definition 2.1 to the case in which the
powers included in the right-hand-side of Eq. (6) are replaced by suitable
distortion functions. In this case, recalling Remark 2.2, we come to an
extension of the Eq. (11).

Let $X$ be a random variabile with CDF $F$ and SF $\overline{F}$, and let
$q_{i}:[0,1]\rightarrow[0,1]$ be two distortion functions, for $i=1,2$, i.e.
increasing functions such that $q_{i}(0)=0$ and $q_{i}(1)=1$ (cf. Section
2.9.2 of Belzunce et al. [4] and Section 2.4 of Navarro [37]). In some
applications it is required that $q_{i}$ is continuous or left-continuous,
however these assumptions are not necessarily required in general. The
distorted distribution function and the distorted survival function of $F$
through $q_{i}$ are given respectively by

$F_{q_{1}}(x)=q_{1}\left(F(x)\right),\qquad\overline{F}_{q_{2}}(x)=q_{2}\left(\overline{F}(x)\right),\qquad\forall
x\in\mathbb{R}.$ (36)

From Eq. (36), in general one has $F_{q_{1}}(x)+\overline{F}_{q_{2}}(x)\neq
1$, for all $x\in(l,r)$, unless $q_{1}(u)=1-q_{2}(1-u)$. The functions given
in (36) have been introduced in the context of the theory of choice under risk
(see Wang [48]), and are largely used in various applied fields (for instance,
see Sordo and Suárez-Llorens [45] for applications to variability measures).

Let us now consider the preannounced generalization of the CIGF based on (36).

###### Definition 5.1

Let $X$ be a random variable with CDF $F(x)$ and SF $\overline{F}(x)$,
$x\in\mathbb{R}$, and let

$l=\inf\\{x\in\mathbb{R}:F(x)>0\\},\qquad
r=\sup\\{x\in\mathbb{R}:\overline{F}(x)>0\\}.$

The q-distorted Gini function (or, shortly, q-Gini function) of $X$ is defined
as

$\hat{G}_{X}({\bf
q})=\int_{l}^{r}F_{q_{1}}(x)\overline{F}_{q_{2}}(x)\,{\rm{d}}x,$ (37)

where ${\bf q}=(q_{1},q_{2})$, and $q_{i}:[0,1]\rightarrow[0,1]$, $i=1,2$, are
distortion functions such that $\hat{G}_{X}({\bf q})$ is finite.

Clearly, if the distortion functions are taken as $q_{1}(x)=x^{\alpha}$ and
$q_{2}(x)=x^{\beta}$ with $(\alpha,\beta)\in D_{X}$, then Eq. (37) corresponds
to the definition of the CIGF given in Eq. (6). Specifically, if
$\alpha=\beta=1$ then we recover the Gini mean semi-difference (11).

It is worth mentioning that the q-Gini function may be viewed as an extension
of the distorted measures treated in Giovagnoli and Wynn [20] and in Greselin
and Zitikis [23]. In these papers, distortions of the only CDF or SF (which
can be viewed as generalizations of Eqs. (12) and (13)), are considered for
the analysis of stochastic dominance, Lorenz ordering and risk measures.

Hereafter we shall prove various results regarding the q-Gini function. They
include the effect of an affine transformation of $X$ and the pointwise
ordering of the q-Gini functions. To this aim, we recall that, if $X$ and $Y$
are nonnegative random variables with CDFs $F$ and $G$, respectively, then $X$
is said to be smaller than $Y$ in the dispersive order, denoted as
$X\leq_{d}Y$, if and only if (see Section 3.B of Shaked and Shanthikumar [43])

$F^{-1}(v)-F^{-1}(u)\geq G^{-1}(v)-G^{-1}(u)\qquad\hbox{whenever }0<u\leq
v<1.$

Among the variability stochastic orders, the dispersive order is one of the
most popular, since it involves quantities that are easily tractable,
requiring that the difference between any two quantiles of $X$ is smaller than
the corresponding quantity of $Y$. Moreover, if $X$ and $Y$ are absolutely
continuous with PDFs $f$ and $g$, respectively, then

$X\leq_{d}Y\quad\hbox{if and only if}\quad f(F^{-1}(u))\geq
g(G^{-1}(u))\quad\forall\,u\in(0,1).$ (38)

We can now prove that, under suitable assumptions, the q-Gini function is a
variability measure in the sense of Bickel and Lehmann [6].

###### Theorem 5.1

Let $X$ and $Y$ be random variables having the same support, and let ${\bf
q}=(q_{1},q_{2})$, where $q_{i}:[0,1]\rightarrow[0,1]$, $i=1,2$, be distortion
functions such that $\hat{G}_{X}({\bf q})$ and $\hat{G}_{Y}({\bf q})$ are
finite. Then, the following properties hold:

  1. 1.

$\hat{G}_{X+\delta}({\bf q})=\hat{G}_{X}({\bf q})$ for all
$\delta\in\mathbb{R}$;

  2. 2.

$\hat{G}_{\gamma X}({\bf q})=\gamma\hat{G}_{X}({\bf q})$ for all
$\gamma\in\mathbb{R}^{+}$;

  3. 3.

$\hat{G}_{X}({\bf q})=0$ for any degenerate random variable $X$;

  4. 4.

$\hat{G}_{X}({\bf q})\geq 0$ for any random variable $X$;

  5. 5.

$X\leq_{d}Y$ implies $\hat{G}_{X}({\bf q})\leq\hat{G}_{Y}({\bf q})$.

  * Proof.

The properties 1 and 2 follow recalling Eq. (37) and the relation between the
CDFs of $X$ and $Y=\gamma X+\delta$. The properties 3 and 4 are guaranteed by
the Definition 5.1. In analogy to Eq. (10), we can write

$\hat{G}_{Y}({\bf q})-\hat{G}_{X}({\bf
q})=\int_{0}^{1}q_{1}(u)q_{2}(1-u)\left[\frac{1}{g(G^{-1}(u))}-\frac{1}{f(F^{-1}(u))}\right]\,\text{d}u.$

Hence, due to relation (38) the property 5 immediately follows.

$\Box$

In addition, we remark that if $\hat{G}_{X}({\bf q})=0$ and if
$q_{i}(0^{+})>0$ and $q_{i}(1^{-})<1$ for $i=1,2$, then $X$ is necessarily a
degenerate random variable.

It is worth mentioning that the results given in this section can be further
extended. Indeed, in the same conditions given in the Definition 5.1, by
taking as reference the approach in [20] we introduce the weighted q-distorted
Gini function (or, shortly, weighted q-Gini function) as follows:

$\hat{G}_{X}({\bf
q},F_{T})=\int_{\Delta}F_{q_{1}}(x)\overline{F}_{q_{2}}(x)\,{\rm{d}}F_{T}(x),$
(39)

where $F_{T}(x)$ is the CDF of a random variable $T$, and where the
intersection of the supports of $X$ and $T$ is a non-empty set denoted by
$\Delta$.

It is not hard to see that the function given in (39) satisfies the properties
1–4 given in Theorem 5.1 for $\hat{G}_{X}({\bf q})$. Concerning the property
5, hereafter we see that additional assumptions are needed. Here, $l_{X}$,
$r_{X}$ and $l_{Y}$, $r_{Y}$ are defined as in (5) for $X$ and $Y$,
respectively.

###### Theorem 5.2

Let $X$ and $Y$ be random variables having the same support, let ${\bf
q}=(q_{1},q_{2})$, where $q_{i}:[0,1]\rightarrow[0,1]$, $i=1,2$, be distortion
functions such that $\hat{G}_{X}({\bf q},F_{T})$ and $\hat{G}_{Y}({\bf
q},F_{T})$ are finite, and let $T$ be absolutely continuous with PDF $f_{T}$.
If
(i) $f_{T}(t)$ is increasing in $t$ and $-\infty<l_{X}=l_{Y}$, or if
(ii) $f_{T}(t)$ is decreasing in $t$ and $r_{X}=r_{Y}<\infty$, then

$X\leq_{d}Y\quad\text{implies}\quad\hat{G}_{X}({\bf
q},F_{T})\leq\hat{G}_{Y}({\bf q},F_{T}).$ (40)

  * Proof.

Due to (39), by setting $u=F(x)$ one has

$\hat{G}_{Y}({\bf q},F_{T})-\hat{G}_{X}({\bf
q},F_{T})=\int_{0}^{1}q_{1}(u)q_{2}(1-u)\left[\frac{f_{T}(G^{-1}(u))}{g(G^{-1}(u))}-\frac{f_{T}(F^{-1}(u))}{f(F^{-1}(u))}\right]\,\text{d}u.$

Then, under assumption (i), from Theorem 3.B.13 of [43] we have that
assumption $X\leq_{d}Y$ implies $X\leq_{st}Y$, i.e. $F(x)\geq G(x)$ for all
$x\in\mathbb{R}$, so that $G^{-1}(u)\geq F^{-1}(u)$ for all $u\in(0,1)$. The
relation (40) thus follows from (38). The same result can be proved similarly
under assumption (ii).

$\Box$

An immediate application of Theorem 5.2 can be given to the reliability of
multi-component stress-strength systems, as seen in Section 4.1. Consider two
$n$-component systems, the first having i.i.d. strengths
$X_{1},X_{2},\ldots,X_{n}$ distributed as $X$, and the second having i.i.d.
strengths $Y_{1},Y_{2},\ldots,Y_{n}$ distributed as $Y$. Assume that each
component of both systems is stressed according to an independent random
stress $T$ having CDF $F_{T}(x)$. We denote by $R_{k,n}^{X}$ and $R_{k,n}^{Y}$
the reliability of the corresponding multi-component stress-strength systems
defined as in (33). We are now able to provide a comparison result based on
the weighted q-Gini function.

###### Theorem 5.3

Let the strengths $X$ and $Y$ have the same support, and let the random stress
$T$ be absolutely continuous with PDF $f_{T}$. If
(i) $f_{T}(t)$ is increasing in $t$ and $-\infty<l_{X}=l_{Y}$, or if
(ii) $f_{T}(t)$ is decreasing in $t$ and $r_{X}=r_{Y}<\infty$, then

$X\leq_{d}Y\quad\text{implies}\quad R_{k,n}^{X}\leq R_{k,n}^{Y}\quad\text{for
all}\quad 0\leq k\leq n,$

provided that $R_{k,n}^{X}$ and $R_{k,n}^{Y}$ are finite.

  * Proof.

The thesis follows recalling Eq. (33) and making use of Theorem 5.2 when the
relevant distortions are given by $q_{1}(u)=u^{n-j}$ and $q_{2}(u)=u^{j}$,
with $k\leq j\leq n$.

$\Box$

In the last result of this section, thanks to the probabilistic analogue of
the mean value theorem, we provide a suitable expression of the weighted
q-Gini function in the special case when the related distortion functions are
equal to the identity.

###### Proposition 5.1

Let $X$ be a nondegenerate random variable such that
$\mathbb{E}(\min\\{X,X^{\prime}\\})$ and $\mathbb{E}(\max\\{X,X^{\prime}\\})$
are finite, where $X^{\prime}$ is an independent copy of $X$. For the weighted
q-Gini function introduced in Eq. (39), if $T$ is absolutely continuous with
the same support of $X$, and if $q_{i}(u)=u$, $0\leq u\leq 1$, for $i=1,2$,
then

$\hat{G}_{X}({\bf
q},F_{T})=\frac{1}{2}\,\mathbb{E}\left[F_{T}(\max\\{X,X^{\prime}\\})-F_{T}(\min\\{X,X^{\prime}\\})\right].$
(41)

  * Proof.

The proof follows making use of Eq. (39) and Theorem 4.1 in Di Crescenzo [13]
extended to the case of a general support of $X$, by taking
$Z=\Psi(\min\\{X,X^{\prime}\\},\max\\{X,X^{\prime}\\})$ and
$g(\cdot)=F_{T}(\cdot)$.

$\Box$

We immediately note that Eq. (41) generalizes Eq. (11) in this proposed
context.

##  6 Two-dimensional cumulative information generating function

Let us now extend the analysis of the CIGF to the case of a two-dimensional
random vector. In analogy with Definition 2.1, by avoiding trivial degenerate
cases, we introduce the following

###### Definition 6.1

Let $(X,Y)$ be random vector with nondegenerate components, having joint CDF
and SF given respectively by

$F(x,y)=P(X\leq x,Y\leq
y),\qquad\overline{F}(x,y)=P(X>x,Y>y),\qquad(x,y)\in\mathbb{R}^{2}.$

We consider the following domain

$\mathcal{S}_{(X,Y)}=\\{(x,y)\in\mathbb{R}^{2}:F(x,y)\overline{F}(x,y)>0\\}.$

The CIGF of $(X,Y)$ is defined as:

$\displaystyle G_{(X,Y)}:$
$\displaystyle\>D_{(X,Y)}\subseteq\mathbb{R}^{2}\longrightarrow(0,+\infty)$
$\displaystyle\qquad(\alpha,\beta)\;\quad\;\longmapsto\;G_{(X,Y)}(\alpha,\beta)=\iint_{\mathcal{S}_{(X,Y)}}[F(x,y)]^{\alpha}\>[\overline{F}(x,y)]^{\beta}\,{\rm{d}}x\,{\rm{d}}y$

where

$D_{(X,Y)}=\\{(\alpha,\beta)\in\mathbb{R}^{2}:G_{(X,Y)}(\alpha,\beta)<+\infty\\}.$

We first discuss few examples. The first example is stimulated by the fact
that if $X\sim\text{Bernoulli}\left(\frac{1}{2}\right)$, then
$G_{X}(\alpha,\beta)=\left(\frac{1}{2}\right)^{\alpha+\beta}$ (cf. Table 2),
thus satisfying the symmetry conditions expressed in Remark 2.3.

###### Example 6.1

Let $(X,Y)$ be a discrete random vector, with probability function

$\mathbb{P}(X=x,Y=y)=\begin{cases}\frac{1}{4}+\theta\qquad(x,y)\in\\{(0,0),(1,1)\\}\\\
\frac{1}{4}-\theta\qquad(x,y)\in\\{(0,1),(1,0)\\}\\\
0\qquad\qquad\text{otherwise}\end{cases}$

for $\theta\in\left(-\frac{1}{4},\frac{1}{4}\right)$. Therefore the CDF and
the SF are identical, given by $F(x,y)=\overline{F}(x,y)=\frac{1}{4}+\theta$,
for $(x,y)\in\mathcal{S}_{(X,Y)}=[0,1]^{2}$. Hence, from Definition 6.1 the
CIGF of $(X,Y)$ is

$G_{(X,Y)}(\alpha,\beta)=\left(\frac{1}{4}+\theta\right)^{\alpha+\beta},\qquad(\alpha,\beta)\in
D_{(X,Y)}=\mathbb{R}^{2}.$

###### Example 6.2

Let $(X,Y)$ be an absolutely continuous random vector, uniformly distributed
in the triangular domain $\mathcal{T}=\left\\{(x,y)\in\mathbb{R}^{2}:0\leq
x\leq 1,\,0\leq y\leq 1-x\right\\}$. The PDF, CDF and SF are given
respectively by

$f(x,y)=2,\qquad
F(x,y)=2xy,\qquad\overline{F}(x,y)=(1-x-y)^{2},\qquad\hbox{for
$(x,y)\in\mathcal{T}$.}$

In this case $\mathcal{S}_{(X,Y)}=\mathcal{T}$, so that the CIGF of $(X,Y)$ is

$\displaystyle G_{(X,Y)}(\alpha,\beta)$
$\displaystyle=\int_{0}^{1}\int_{0}^{1-x}\left(2xy\right)^{\alpha}\>\left(1-x-y\right)^{2\beta}\>{\rm{d}}y\,{\rm{d}}x$
$\displaystyle=2^{\alpha}B(\alpha+1,2\beta+1)\int_{0}^{1}x^{\alpha}(1-x)^{\alpha+2\beta+1}\,{\rm{d}}x$
$\displaystyle=2^{\alpha}B(\alpha+1,2\beta+1)B(\alpha+1,\alpha+2\beta+2),$

with

$D_{(X,Y)}=\left\\{(\alpha,\beta)\in\mathbb{R}^{2}:\alpha>-1,\beta>-\frac{1}{2}\right\\}.$

###### Example 6.3

Let $(X,Y)$ be an absolutely continuous random vector, distributed on the
domain $\mathcal{Q}=[0,1]^{2}$, with PDF, CDF and SF given respectively by,
for $(x,y)\in\mathcal{Q}$,

$f(x,y)=x+y,\quad
F(x,y)=\frac{1}{2}xy(x+y),\quad\overline{F}(x,y)=\frac{1}{2}(x-1)(y-1)(x+y+2),$

so that in this case $\mathcal{S}_{(X,Y)}=\mathcal{Q}$. Since
$\left|\frac{x+y}{2}\right|<1$ for $(x,y)\in\mathcal{Q}$, recalling the
expression of the Gauss hypergeometric function (cf. 15.3.1 of Abramowitz and
Stegun [2])

${}_{2}{F_{1}(a,b,c,z)}=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}\,{\rm{d}}t,\qquad
Re(c)>Re(b)>0,$

and making use of 15.3.7 in [2] and of 2.21.1.4 in Prudnikov et al. [36], the
CIGF of $(X,Y)$ is

$\displaystyle G_{(X,Y)}(\alpha,\beta)$
$\displaystyle=\frac{1}{2^{\alpha}}\sum_{k=0}^{+\infty}{\beta\choose
k}\frac{1}{2^{k}}\frac{\Gamma(\beta+1)\Gamma(2\alpha+k+1)}{\Gamma(2+2\alpha+\beta+k)}B(\alpha+1,\beta+1)$
$\displaystyle\quad\times{}_{3}{F_{2}(-\alpha-k,-1-2\alpha-\beta-k,\alpha+1,-2\alpha-k,-\alpha+\beta+2,-1)}$
$\displaystyle+\frac{1}{2^{\alpha}}\sum_{k=0}^{+\infty}{\beta\choose
k}\frac{1}{2^{k}}\frac{\Gamma(\alpha+1)\Gamma(-2\alpha-k-1)}{\Gamma(-\alpha-k)}B(3\alpha+k+2,\beta+1)$
$\displaystyle\quad\times{}_{3}{F_{2}(\alpha+1,-\beta,3\alpha+k+2,2+2\alpha+k,3\alpha+\beta+3,-1)}$

with
$D_{(X,Y)}=\\{(\alpha,\beta)\in\mathbb{R}^{2}:\alpha,\beta\in\mathbb{R}\setminus\mathbb{Z}_{0}^{-}\\}$,
where the function ${}_{3}F_{2}$ can be found in [36], for instance.

The following result is an immediate consequence of the involved notions.

###### Proposition 6.1

Let $(X,Y)$ be a random vector having finite CIGF. If $X$ and $Y$ are
independent then

$G_{(X,Y)}(\alpha,\beta)=G_{X}(\alpha,\beta)\>G_{Y}(\alpha,\beta)\qquad\forall\,(\alpha,\beta)\in
D_{(X,Y)}.$

Consider a nonnegative random vector $(X,Y)$ with support
$(0,r_{1})\times(0,r_{2})$, for $r_{1},r_{2}\in(0,+\infty]$. In many practical
situations, it is worthwhile to adopt the following information measures for
multi-device systems. The _joint cumulative residual entropy_ of $(X,Y)$ is
defined as (cf. [41])

$\mathcal{CRE}(X,Y)=-\int_{0}^{r_{1}}\text{d}x\int_{0}^{r_{2}}\overline{F}(x,y)\log\overline{F}(x,y)\,\text{d}y.$
(42)

A dynamic version of this measure has been studied by Rajesh et al. [39].
Similarly, the _joint cumulative entropy_ of $(X,Y)$ is defined as (cf. [17])

$\mathcal{CE}(X,Y)=-\int_{0}^{r_{1}}\text{d}x\int_{0}^{r_{2}}F(x,y)\log
F(x,y)\,\text{d}y.$ (43)

###### Remark 6.1

We recall that, if $(X,Y)$ has support $(0,r_{1})\times(0,r_{2})$, for
$r_{1},r_{2}\in(0,+\infty)$, and if $X$ and $Y$ are independent, then (cf.
Proposition 2.2 of [17])

$\mathcal{CE}(X,Y)=\left[r_{2}-\mathbb{E}(Y)\right]\mathcal{CE}(X)+\left[r_{1}-\mathbb{E}(X)\right]\mathcal{CE}(Y).$
(44)

Under the same assumptions, similarly, from Eq. (42) we can observe that

$\mathcal{CRE}(X,Y)=\mathbb{E}\left(Y\right)\mathcal{CRE}(X)+\mathbb{E}\left(X\right)\mathcal{CRE}(Y).$
(45)

###### Remark 6.2

For the discrete random vector $(X,Y)$ considered in Example 6.1, we have

$\mathcal{CRE}(X)=\mathcal{CRE}(Y)=\mathcal{CE}(X)=\mathcal{CE}(Y)=-\frac{1}{2}\log\left(\frac{1}{2}\right)$

and

$\mathcal{CRE}(X,Y)=\mathcal{CE}(X,Y)=-\left(\frac{1}{4}+\theta\right)\log\left(\frac{1}{4}+\theta\right),\qquad\theta\in\left(-\frac{1}{4},\frac{1}{4}\right).$

Hence, in this case Eqs. (44) and (45) are satisfied if and only if $X$ and
$Y$ are independent, i.e. $\theta=0$.

In analogy with the one-dimensional measures considered in Table 1, we define
the generalized and the fractional versions of the measures given in Eqs. (42)
and (43).

###### Definition 6.2

Let $(X,Y)$ be a nonnegative random vector with support
$(0,r_{1})\times(0,r_{2})$, where $r_{1},r_{2}\in(0,+\infty]$. The generalized
cumulative residual entropy of order $n$ of $(X,Y)$ is defined as

$\mathcal{CRE}_{n}(X,Y)=\frac{1}{n!}\int_{0}^{r_{1}}{\rm{d}}x\int_{0}^{r_{2}}\overline{F}(x,y)\left[-\log\overline{F}(x,y)\right]^{n}{\rm{d}}y,\qquad
n\in\mathbb{N}_{0},$

while the generalized cumulative entropy of order $n$ of $(X,Y)$ is defined as

$\mathcal{CE}_{n}(X,Y)=\frac{1}{n!}\int_{0}^{r_{1}}{\rm{d}}x\int_{0}^{r_{2}}F(x,y)\left[-\log
F(x,y)\right]^{n}{\rm{d}}y,\qquad n\in\mathbb{N}.$

###### Definition 6.3

Let $(X,Y)$ be a nonnegative random vector with support
$(0,r_{1})\times(0,r_{2})$, with $r_{1},r_{2}\in(0,+\infty]$. The fractional
cumulative residual entropy of $(X,Y)$ is defined as

$\mathcal{CRE}_{\nu}(X,Y)=\frac{1}{\Gamma(\nu+1)}\int_{0}^{r_{1}}{\rm{d}}x\int_{0}^{r_{2}}\overline{F}(x,y)\left[-\log\overline{F}(x,y)\right]^{\nu}{\rm{d}}y,\qquad\nu\geq
0$

whereas the fractional cumulative entropy of $(X,Y)$ is defined as

$\mathcal{CE}_{\nu}(X,Y)=\frac{1}{\Gamma(\nu+1)}\int_{0}^{r_{1}}{\rm{d}}x\int_{0}^{r_{2}}F(x,y)\left[-\log
F(x,y)\right]^{\nu}{\rm{d}}y,\qquad\nu>0.$

According to Propositions 2.1, 2.2 and 2.3, now we propose the following
generalizations, whose proof is analogous.

###### Proposition 6.2

Let $(X,Y)$ be a random vector with finite CIGF $G_{(X,Y)}(\alpha,\beta)$, for
$(\alpha,\beta)\in D_{(X,Y)}$, and let
${\mathcal{S}_{(X,Y)}}=(0,r_{1})\times(0,r_{2})$ where
$r_{1},r_{2}\in(0,+\infty]$. If $(0,1)\in D_{(X,Y)}$, then

$\mathcal{CRE}(X,Y)=-\frac{\partial}{\partial\beta}G_{(X,Y)}(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1},$
$\mathcal{CRE}_{n}(X,Y)=\frac{(-1)^{n}}{n!}\frac{\partial^{n}}{\partial\beta^{n}}G_{(X,Y)}(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1},\qquad
n\in\mathbb{N}_{0},$
$\mathcal{CRE}_{\nu}(X,Y)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-,\beta}^{\nu}G_{(X,Y)}}\right)(\alpha,\beta)\Big{|}_{\alpha=0,\beta=1},\qquad\nu>0,$

If $(1,0)\in D_{(X,Y)}$, then

$\mathcal{CE}(X,Y)=-\frac{\partial}{\partial\alpha}G_{(X,Y)}(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0},$
$\mathcal{CE}_{n}(X,Y)=\frac{(-1)^{n}}{n!}\frac{\partial^{n}}{\partial\alpha^{n}}G_{(X,Y)}(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0},\qquad
n\in\mathbb{N},$
$\mathcal{CE}_{\nu}(X,Y)=\frac{1}{\Gamma(\nu+1)}\left(\prescript{C}{}{D_{-,\alpha}^{\nu}G_{(X,Y)}}\right)(\alpha,\beta)\Big{|}_{\alpha=1,\beta=0},\qquad\nu>0.$

In analogy with Eq. (21), if $(X,Y)$ is a random vector with CDF $F(x,y)$ and
SF $\overline{F}(x,y)$, for all $(x,y)\in\mathcal{S}_{(X,Y)}$ we can introduce
the two-dimensional _odds function_ as

$\theta(x,y)=\frac{\overline{F}(x,y)}{F(x,y)}.$ (46)

Hence, the two-dimensional extension of Eq. (22) can be given in terms of the
function in (46), so that

$G_{(X,Y)}(-\beta,\beta)=\iint_{\mathcal{S}_{(X,Y)}}[\theta(x,y)]^{\beta}\>\text{d}x\>\text{d}y$

for all $\beta\in\mathbb{R}$ such that the right-hand-side is finite.

##  7 Concluding remarks

In this paper, we defined the cumulative information generating function and
its suitable distortions-based extensions. The CIGF is noteworthy in the
context of information measures, since it allows to determine the cumulative
residual entropy and the cumulative entropy of a given probability
distribution, even in their generalized and fractional forms.

Several results, properties and bounds have been studied, also with reference
to symmetry properties and relations with the equilibrium density. Moreover,
we illustrated that the considered functions are variability measures which
extend the Gini mean semi-difference.

In the realm of reliability theory, the CIGF and its extensions have been
found useful to study properties of $k$-out-of-$n$ systems, with special
regards to the reliability of multi-component stress-strength systems.

Future developments can be oriented to connections with other notions, such as
the Information Divergence (see Toulias and Kitsos [47]) and the Rényi Entropy
(see, for instance, Buryak and Mishura [8]), also with reference to possible
applications and distortions-based extensions of the two-dimensional CIGF
introduced in Section 6.

## Credit Author Statement

All authors have contributed equally to this work.

## 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

The authors are members of the group GNCS of INdAM (Istituto Nazionale di Alta
Matematica). This work is partially supported by MIUR–PRIN 2017, Project
“Stochastic Models for Complex Systems” (no. 2017JFFHSH).

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