# Different informational characteristics of cubic transmuted distributions**

Shital Saha1, Suchandan Kayal2, and N. Balakrishnan3 Email address:
<EMAIL_ADDRESS>author : Suchandan Kayal
<EMAIL_ADDRESS>suchandan.kayal@gmail.com)Email address:
<EMAIL_ADDRESS>
**It has been accepted on Brazilian Journal of Probability and Statistics.

1,2Department of Mathematics, National Institute of Technology Rourkela,
Rourkela-769008, Odisha, India
3Department of Mathematics and Statistics, McMaster University, Hamilton,
Ontario L8S 4K1, Canada

Abstract

Cubic transmuted (CT) distributions were introduced recently by Granzotto et
al. (2017). In this article, we derive Shannon entropy, Gini’s mean difference
and Fisher information (matrix) for CT distributions and establish some of
their theoretical properties. In addition, we propose cubic transmuted Shannon
entropy and cubic transmuted Gini’s mean difference. The CT Shannon entropy is
expressed in terms of Kullback-Leibler divergences, while the CT Gini’s mean
difference is shown to be connected with energy distances. We show that the
Kullback-Leibler and Chi-square divergences are free of the underlying parent
distribution. Finally, we carry out some simulation studies for the proposed
information measures from an inferential viewpoint.

Keywords. Quadratic transmuted distributions; CT Shannon entropy; CT Gini’s
mean difference; Fisher information; Kullback-Leibler divergence; Chi-square
divergence; Cubic transmuted distributions.

2020 Mathematics Subject Classification. Primary 62B10, 81P45.

##  1 Introduction

Recently, many different generalized families of distributions have been
proposed in the literature in order to provide greater flexibility while
modelling data arising from diverse applications. In this regard, Shaw and
Buckley (2009) introduced an important class of distributions, known as the
class of transmuted distributions, by using a quadratic ranking transmutation
map. Let $F(\cdot)$ be the cumulative distribution function (CDF) of a
baseline distribution. Then, the CDF of the transmuted distribution is (Shaw
and Buckley (2009))

$\displaystyle F_{X_{T}}(x)=(1+\lambda)F(x)-\lambda
F^{2}(x),~{}~{}|\lambda|\leq 1~{}~{}x\in\mathbb{R},$ (1.1)

where $\mathbb{R}$ denotes the set of real numbers. The transmuted generalized
extreme value distribution was studied by Aryal and Tsokos (2009). They also
described an application of the transmuted Gumbel distribution for the climate
data. There are some real life data sets whose complexity can not be
adequately captured using quadratic transmuted distributions. A significant
amount of work has been made towards developing a new transmuted model and
subsequently discussing its enhanced flexibility in modelling various types of
real life data, where the parent model does not provide a good fit. Granzotto
et al. (2017) established a mixture representation for the transmuted
distribution in (1.1), using distributions of order statistics. To be
specific, with $X_{1}$ and $X_{2}$ being i.i.d. random variables with
distribution function $F(x)$ and $(X_{1:2},X_{2:2})$ denoting the
corresponding order statistics, they considered the mixture random variable

$\displaystyle
X\overset{d}{=}\left\\{\begin{array}[]{ll}X_{1:2}~{}\mbox{with~{}probability}~{}~{}\pi\\\
X_{2:2}~{}\mbox{with~{}probability}~{}1-\pi,\end{array}\right.$ (1.4)

where $0\leq\pi\leq 1$ is the mixing probability. Then, using the CDFs of
$X_{1:2}$ and $X_{2:2}$ given by $1-\\{1-F(x)\\}^{2}$ and $F^{2}(x)$ (see, for
example, Arnold et al. (1992)), respectively, (1.4) readily yields the CDF of
$X$ to be

$\displaystyle F_{X}(x)$ $\displaystyle=$
$\displaystyle\pi\big{[}1-\\{1-F(x)\\}^{2}\big{]}+(1-\pi)F^{2}(x)$ (1.5)
$\displaystyle=$ $\displaystyle 2\pi F(x)+(1-2\pi)F^{2}(x).$

It is evident that the quadratic transmuted distribution in (1.1) is a simple
reparametrization of the mixture distribution in (1.5) (with
$2\pi=1+\lambda$).

Then, proceeding along the same lines and considering $X_{1},X_{2}$ and
$X_{3}$ to be i.i.d. random variables with distribution function $F(x)$ and
$(X_{1:3},X_{2:3},X_{3:3})$ as the corresponding order statistics, Granzotto
et al. (2017) considered the mixture random variable

$\displaystyle
X\overset{d}{=}\left\\{\begin{array}[]{ll}X_{1:3}~{}\mbox{with~{}probability}~{}~{}\pi_{1}\\\
X_{2:3}~{}\mbox{with~{}probability}~{}\pi_{2}\\\
X_{3:3}~{}\mbox{with~{}probability}~{}\pi_{3},\end{array}\right.$ (1.9)

where $\pi_{1},\pi_{2},\pi_{3}\in(0,1)$ are the mixing probabilities such that
$\pi_{1}+\pi_{2}+\pi_{3}=1$. The CDFs and PDFs of $X_{1:3},$ $X_{2:3}$,
$X_{3:3}$ are denoted by $F_{\min}(\cdot)$, $F_{2:3}(\cdot),$
$F_{\max}(\cdot)$ and $f_{\min}(\cdot)$, $f_{2:3}(\cdot),$ $f_{\max}(\cdot)$,
respectively. Then, using the CDFs of $X_{1:3},X_{2:3}$ and $X_{3:3}$ given in
Arnold et al. (1992), for example, (1.9) readily yields the CDF of $X$ to be

$\displaystyle F_{X}(x)$ $\displaystyle=$ $\displaystyle
3\pi_{1}F(x)+3(\pi_{2}-\pi_{1})F^{2}(x)+(1-3\pi_{2})F^{3}(x).$ (1.10)

Upon re-parametrizing the distribution in (1.10) by setting
$3\pi_{1}=\lambda_{1}$ and $3\pi_{2}=\lambda_{2}$, Granzotto et al. (2017)
proposed the cubic transmuted distribution with CDF

$\displaystyle
F_{X_{CT}}(x)=\lambda_{1}F(x)+(\lambda_{2}-\lambda_{1})F^{2}(x)+(1-\lambda_{2})F^{3}(x),$
(1.11)

where $\lambda_{1}\in[0,1]$ and $\lambda_{2}\in[-1,1]$. The density function
corresponding to the CDF in (1.11) is

$\displaystyle
f_{X_{CT}}(x)=f(x)\\{\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\\}.$
(1.12)

The model considered by Rahman et al. (2018) is a simple re-parametrization of
the distribution in (1.11). Observe that if $\lambda_{1}$ is taken to be
$1+\lambda$ and $\lambda_{2}$ to be 1 in (1.11), then the quadratic transmuted
distribution in (1.1) is deduced. Further, if $\lambda_{1}$ is taken to be
$1+\lambda$ and $\lambda_{2}$ to be $1-\lambda$ in (1.11), then the cubic
transmuted distribution with single parameter is deduced (see AL-Kadim and
Mohammed (2017)). The corresponding CDF and PDF of the single parameter CT
distribution are, respectively, given by

$\displaystyle F^{*}_{X_{CT}}(x)=(1+\lambda)F(x)-2\lambda F^{2}(x)+\lambda
F^{3}(x)$ (1.13)

and

$\displaystyle f^{*}_{X_{CT}}(x)=f(x)\left\\{(1+\lambda)-4\lambda
f(x)+3\lambda f^{2}(x)\right\\},$ (1.14)

where $|\lambda|\leq 1$ and $x\in\mathbb{R}$.

Aslam et al. (2018) studied goodness of fit tests that the cubic transmuted
distribution fits better than the parent distributions for the real data set
(see also Rahman et al. (2020) and Ahsan-ul Haq et al. (2023)). Later, Rahman
et al. (2019) showed that cubic transmuted power distribution fits better than
the parent distribution for some real data set. It is important to mention
here that such mixture distributions of order statistics had earlier been
considered in the pioneering work of Lehmann (1953) as a nonparametric
alternative in his development of rank based tests (see also the related
comments by Balakrishnan (2021)).

The uncertainty associated with a probability model can be computed using
several information measures. In many situations related to lifetime data
analysis, experimental physics, econometrics and demography, uncertainty
quantification in a distribution is very important. The uncertainty in a
random variable can be measured using the concepts of Fisher information (see
Fisher (1929)), Shannon entropy (see Shannon (1948)) and Rényi entropy (see
Rényi (1961)). Besides these, various divergence measures such as Kullback-
Leibler (KL) divergence (see Kullback and Leibler (1951)), Chi-square
divergence (see Nielsen and Nock (2013)) have been introduced to measure the
closeness between two probability distributions.

From the above discussion, we note that although many transmuted families of
distributions have been introduced and studied by several researchers, these
models have not been explored from information theoretical viewpoint except
for the work of Kharazmi and Balakrishnan (2021) . Kharazmi and Balakrishnan
(2021) studied transmuted distributions and established several theoretic
informational properties. They also provided extensions of Shannon entropy and
Gini’s mean difference. Motivated by the work of Kharazmi and Balakrishnan
(2021), we extend here their results to the results for the family of cubic
transmuted distributions partially.

In this paper, we explore several informational properties of a more flexible
and general class of cubic transmuted distributions in (1.11). We study
Shannon entropy, Gini’s mean difference, and Fisher information of the class
of cubic transmuted distributions. In addition, we also study KL divergence,
Chi-square divergence, and energy distance divergence (Cramér (1928)) between
CT distribution and its components. We propose extensions of Shannon entropy
and Gini’s mean difference and call these as CT Shannon entropy and CT Gini’s
mean difference, respectively.

The rest of this paper proceeds as follows. In Section $2$, we study Shannon
entropy of a CT distribution. We obtain Kullback-Leibler divergence between CT
distribution and the densities of its components. It is noted that the KL
divergence between a general transmuted model and its components is free of
the parent distribution. The cubic transmuted Shannon entropy is proposed as
an extension of Shannon entropy of CT distribution and Jensen-Shannon (JS)
divergence measure. Further, the proposed CT Shannon entropy is expressed in
terms of KL divergence measures of the general model and its components.
Gini’s mean difference of the CT distributions is proposed and studied in
Section $3.$ Further, Gini’s mean difference of the CT distribution is
expressed as the sum of Gini’s mean difference of the general transmuted
distribution and that of its components. Similar to the CT Shannon entropy, CT
Gini’s mean difference is proposed. In Section $4$, we discuss the Chi-square
divergence between the general transmuted distribution and the density
functions of its components. Section $5$ presents Fisher information for the
CT distribution. We obtain Fisher information matrix for the CT distribution
given in (1.11). In addition, we also present Fisher information of the
particular model with CDF as in (1.13). Section $6$ describes simulation
investigation to show the importance of the proposed information measures from
an inferential viewpoint. Finally, some concluding remarks are added in
Section $7.$

##  2 Shannon entropy, KL divergence and CT Shannon entropy

The purpose of this section is three-fold. First, we study Shannon entropy of
the CT distribution. Then, we derive KL divergences for the general CT
distribution and its components, and finally we define CT Shannon entropy,
which can be treated as an extension of the Shannon entropy and JS divergence.

###  2.1 Shannon entropy

Here, we derive Shannon entropy of the CT distribution with CDF and PDF given
in (1.11) and (1.12), respectively. Let $X$ be an absolutely continuous random
variable with PDF $f(\cdot)$. Then, the Shannon entropy and weighted Shannon
entropy of $X$ are, respectively, given by

$\displaystyle
H(f)=-\int_{-\infty}^{\infty}f(x)\log\big{(}f(x)\big{)}dx~{}~{}\mbox{and}~{}~{}H^{\psi}(f)=-\int_{-\infty}^{\infty}\psi(x)f(x)\log\big{(}f(x)\big{)}dx,$
(2.1)

where $\psi$ is a nonnegative real-valued measurable function. We note that
the concept of weighted entropy takes into account values of different
outcomes and this makes entropy context-dependent, through the choice of
weight function. In particular, when $\psi(x)=x,$ the functional $H^{\psi}(f)$
reduces to a length-biased information measure (see Di Crescenzo and
Longobardi (2006)). A random variable $X$ is said to have beta distribution if
its PDF is of the form

$\displaystyle
g(x)=\frac{x^{a-1}(1-x)^{b-1}}{\mathbb{B}(a,b)},~{}~{}0<x<1,~{}a,b>0,$ (2.2)

where $\mathbb{B}(a,b)$ denotes the complete beta function. Henceforth, we
denote $X\sim Beta(a,b)$ if its PDF is as in (2.2).

######  Theorem 2.1.

Let $X_{1}$, $X_{2},$ and $X_{3}$ be i.i.d. random variables with a common CDF
$F(\cdot)$ and PDF $f(\cdot)$. Then, the Shannon entropy of a CT random
variable $X_{CT}$ is given by

$\displaystyle H(f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\lambda_{1}H(f)+(1-\lambda_{2})H(f_{\max})+2(\lambda_{2}-\lambda_{1})H^{F}(f)-(1-\lambda_{2})H(f_{W})+H(f_{U_{CT}}),$

where $U_{CT}$ is the CT uniform random variable, $W\sim Beta(3,1)$, and
$H^{F}(X)$ means weighted Shannon entropy defined in (2.1) with weight
function $F.$

###### Proof.

The Shannon entropy of the CT random variable $X_{CT}$ is

$\displaystyle
H(f_{X_{CT}})=-\int_{-\infty}^{\infty}f_{X_{CT}}(x)\log\big{(}f_{X_{CT}}(x)\big{)}dx=I_{1}+I_{2},$
(2.3)

where

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\big{[}f(x)\\{\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\\}\big{]}\log\big{(}f(x)\big{)}dx,$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\big{[}f(x)\\{\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\\}\big{]}\log\big{(}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)$
$\displaystyle+~{}3(1-\lambda_{2})F^{2}(x)\big{)}dx.$

Using the transformation $u=F(x)$ in (2.3) and after some algebraic
calculations, we obtain

$\displaystyle H(f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\lambda_{1}H(f)+(1-\lambda_{2})H(f_{\max})+(1-\lambda_{2})\int_{0}^{1}3u^{2}\log(3u^{2})du$
$\displaystyle-\int_{0}^{1}\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+3(1-\lambda_{2})u^{2}\big{\\}}\log\big{(}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u$
$\displaystyle+3(1-\lambda_{2})u^{2}\big{)}du-2(\lambda_{2}-\lambda_{1})\int_{-\infty}^{\infty}F(x)f(x)\log\big{(}f(x)\big{)}dx$
$\displaystyle=$
$\displaystyle\lambda_{1}H(f)+(1-\lambda_{2})H(f_{\max})-(1-\lambda_{2})H(f_{W})+H(f_{U_{CT}})$
$\displaystyle+2(\lambda_{2}-\lambda_{1})H^{F}(f).$

Hence, the result. ∎

Substituting $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1$ in the result of
Theorem 2.1, we obtain

$\displaystyle H(f_{X_{T}})=(1+\lambda)H(f)+H(f_{U_{T}})-2\lambda H^{F}(f),$
(2.4)

where $f_{U_{T}}(\cdot)$ is the PDF of the transmuted uniform random variable.
Now,

$\displaystyle-2\lambda H^{F}(f)$ $\displaystyle=$ $\displaystyle
2\lambda\int_{-\infty}^{\infty}F(x)f(x)\log\big{(}f(x)\big{)}dx$ (2.5)
$\displaystyle=$ $\displaystyle
2\lambda_{1}\int_{-\infty}^{\infty}F(x)f(x)\log\Big{(}\frac{2F(x)f(x)}{2F(x)}\Big{)}dx$
$\displaystyle=$
$\displaystyle\lambda\int_{-\infty}^{\infty}2F(x)f(x)\log\big{(}2F(x)f(x)\big{)}dx-\lambda\int_{-\infty}^{\infty}2F(x)f(x)\log\big{(}2F(x)\big{)}dx$
$\displaystyle=$ $\displaystyle-\lambda
H(f_{\max})-\lambda\int_{0}^{1}2u\log(2u)du~{}~{}\big{(}\mbox{using}~{}u=F(x)\big{)}$
$\displaystyle=$ $\displaystyle-\lambda H(f_{\max})+\lambda H(f_{V}),$

where $f_{V}(\cdot)$ is the PDF of $V\sim Beta(2,1).$ In the following remark,
we study Shannon entropy for two special cases of CT distribution.

######  Remark 2.1.

  * •

For $\lambda_{2}=1$ and $\lambda_{1}=1+\lambda$, using (2.5), the Shannon
entropy of CT distribution becomes

$\displaystyle H(f_{X_{T}})=(1+\lambda)H(f)-\lambda H(f_{\max})+\lambda
H(f_{V})+H(f_{U_{T}}),$

where $U_{T}$ is the transmuted uniform random variable and $V\sim Beta(2,1)$.
For the details, one may refer to Kharazmi and Balakrishnan (2021).

  * •

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1-\lambda$, the expression of
Shannon entropy for the CT distribution in Theorem 2.1 reduces to the Shannon
entropy of one parameter CT distribution with CDF in (1.13), which is given by

$\displaystyle H(f^{*}_{X_{CT}})=(1+\lambda)H(f)+\lambda H(f_{\max})-2\lambda
H^{F}(f)-\lambda H(f_{W})+H(f^{*}_{U_{CT}}).$

######  Remark 2.2.

Define
$\theta(\lambda_{1},\lambda_{2})=H(f_{U_{CT}})-(1-\lambda_{2})H(f_{W})$. Then,
after some algebraic calculations, we obtain a closed-form expression for
$\theta(\lambda_{1},\lambda_{2})$ as

$\displaystyle\theta(\lambda_{1},\lambda_{2})$ $\displaystyle=$
$\displaystyle-\Big{\\{}\log(3-\lambda_{1}-\lambda_{2})-\phi(p)+\phi(q)-\frac{2}{3}(3+\lambda_{1}-4\lambda_{2})$
$\displaystyle+\frac{2}{9(1-\lambda_{2})}(\lambda^{2}_{1}+13\lambda^{2}_{2}-2\lambda_{1}\lambda_{2}-18\lambda_{2}+6)\Big{\\}}-(1-\lambda_{2})\Big{\\{}\frac{2}{3}-\log(3)\Big{\\}},$

where

$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\left\\{2(\lambda_{2}-\lambda_{1})^{2}x^{2}+10(\lambda_{2}-\lambda_{1})(1-\lambda_{2})x^{3}+12(1-\lambda_{2})^{2}x^{4}\right\\}\log\Big{(}\frac{x}{x-1}\Big{)},$
$\displaystyle r$ $\displaystyle=$
$\displaystyle\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{1}\lambda_{2}-3\lambda_{1}},~{}p=\frac{\lambda_{1}-\lambda_{2}+r}{3(1-\lambda_{2})},~{}q=\frac{\lambda_{1}-\lambda_{2}-r}{3(1-\lambda_{2})}.$

Because of the complex about its form of $\theta(\lambda_{1},\lambda_{2})$, it
is difficult to analyse it from a theoretical viewpoint. To get an idea about
its codomain, we plot the function with respect to $\lambda_{1}$ when
$\lambda_{2}$ is fixed, and with respect to $\lambda_{2}$ when $\lambda_{1}$
is fixed (see Figure 1). From the figure, it is clearly seen that
$\theta(\lambda_{1},\lambda_{2})$ takes both positive and negative values.

Figure 1: Plots of $\theta(\lambda_{1},\lambda_{2})$ for different choices of
the parameters.

Next, we consider the following example to illustrate the result in Theorem
2.1.

######  Example 2.1.

Let $X_{1}$, $X_{2},$ and $X_{3}$ be i.i.d. Pareto variables with PDF
$f(x)=1-x^{-\alpha},~{}x\geq 1,~{}\alpha>0$. Then, the Shannon entropy for the
CT random variable $X_{CT}$ is given by

$\displaystyle H(f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\lambda_{1}\Big{\\{}\frac{\alpha+1}{\alpha^{2}}-\log(\alpha)\Big{\\}}+(1-\lambda_{2})\Big{\\{}\frac{1}{6}-\frac{7}{6\alpha}-\log(3\alpha)\Big{\\}}$
$\displaystyle+(\lambda_{2}-\lambda_{1})\Big{\\{}\frac{1}{2\alpha}(\alpha+1)(3-2\alpha)-\log(\alpha)\Big{\\}}+\theta(\lambda_{1},\lambda_{2}),$

where $\theta(\lambda_{1},\lambda_{2})$ is as given in Remark 2.2.

######  Example 2.2.

Let $X_{1}$, $X_{2}$ and $X_{3}$ be i.i.d. exponential variables with PDF
$f(x)=\beta e^{-\beta x}$, $x>0,$ $\beta>0$. Then, the Shannon entropy of
$X_{CT}$ is given by

$\displaystyle
H(f_{X_{CT}})=\lambda_{1}\big{\\{}1-\log(\beta)\big{\\}}+(1-\lambda_{2})\Big{\\{}\log(3\beta)-\frac{5}{2}\Big{\\}}-(\lambda_{2}-\lambda_{1})\log(\beta)+\theta(\lambda_{1},\lambda_{2}),$

where $\theta(\lambda_{1},\lambda_{2})$ is as given in Remark 2.2.

###  2.2 Kullback-Leibler divergence

We prove the following result which shows that Kullback-Leibler divergence
measures between CT distribution and its components are equal to those between
CT uniform distribution and its components. Let $f_{1}(\cdot)$ and
$f_{2}(\cdot)$ be the PDFs of two absolutely continuous random variables, say
$X_{1}$ and $X_{2}$, respectively. Then, the Kullback-Leibler divergence
between $f_{1}(\cdot)$ and $f_{2}(\cdot)$ is given by (see Kullback and
Leibler (1951))

$\displaystyle
KL(f_{1},f_{2})=\int_{-\infty}^{\infty}f_{1}(x)\log\left(\frac{f_{1}(x)}{f_{2}(x)}\right)dx.$
(2.6)

Note that the Kullback-Leibler divergence is not symmetric, that is, in
general, $KL(f_{1},f_{2})\neq KL(f_{2},f_{1})$. Further, it can be established
that Kullback-Leibler divergence takes non-negative values, and that it could
it may be infinity.

######  Theorem 2.2.

Suppose $X$ is a random variable with PDF $f(\cdot)$. Further, let the CT
random variable $X_{CT}$ have PDF $f_{X_{CT}}(\cdot)$. Then,

  * (a)

$KL(f,f_{X_{CT}})=KL(f_{U},f_{U_{CT}}),$

  * (b)

$KL(f_{X_{CT}},f)=KL(f_{U_{CT}},f_{U}),$

where uniform and CT uniform random variables, denoted by $U$ and $U_{CT}$ in
the interval $(0,1)$, have PDFs $f_{U}(\cdot)$ and $f_{U_{CT}}(\cdot)$,
respectively.

###### Proof.

$(a)$ The Kullback-Leibler divergence between $f(\cdot)$ and
$f_{X_{CT}}(\cdot)$ is given by

$\displaystyle KL(f,f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f(x)\log\bigg{(}\dfrac{f(x)}{f_{X_{CT}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f(x)\log\bigg{(}\dfrac{f(x)}{f(x)\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\big{\\}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}\log\big{(}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+3(1-\lambda_{2})u^{2}\big{)}du~{}\big{(}\mbox{using}~{}u=F(x)\big{)}$
$\displaystyle=$ $\displaystyle KL(f_{U},f_{U_{CT}}).$

$(b)$ The second part can be proved similarly to the first part, and is
therefore omitted. ∎

######  Theorem 2.3.

Suppose $X_{1}$, $X_{2}$ and $X_{3}$ are i.i.d. as $X$ with PDF $f(\cdot)$.
Further, let $f_{2:3}(\cdot)$ be the PDF of $X_{2:3}$. Then,

  * (a)

$KL(f_{2:3},f_{X_{CT}})=KL(f_{U_{2:3}},f_{U_{CT}}),$

  * (b)

$KL(f_{2:3},f_{\max})=KL(f_{U_{2:3}},f_{W}),$

where $f_{W}(\cdot)$ is the PDF of $W\sim Beta(3,1)$, $U_{CT}$ is as defined
in Theorem 2.2, and $f_{U_{2:3}}(\cdot)$ is the PDF of $U_{2:3}$.

###### Proof.

$(a)$ The Kullback-Leibler divergence between $f_{2:3}(\cdot)$ and
$f_{X_{CT}}(\cdot)$ is given by

$\displaystyle KL(f_{2:3},f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f_{2:3}(x)\log\bigg{(}\dfrac{f_{2:3}(x)}{f_{X_{CT}}(x)}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f_{2:3}(x)\log\bigg{(}\dfrac{6f(x)\\{F(x)-F^{2}(x)\\}}{f(x)\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\big{\\}}}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{1}6(u-u^{2})\log\bigg{(}\dfrac{6(u-u^{2})}{\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+(1-\lambda_{2})u^{2}}\bigg{)}du~{}\big{(}\mbox{using}~{}u=F(x)\big{)}$
$\displaystyle=$ $\displaystyle KL(f_{U_{2:3}},f_{U_{CT}}).$

$(b)$ The Kullback-Leibler divergence between $f_{2:3}(\cdot)$ and
$f_{\max}(\cdot)$ is

$\displaystyle KL(f_{2:3},f_{\max})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f_{2:3}(x)\log\bigg{(}\dfrac{f_{2:3}(x)}{f_{\max}(x)}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}6f(x)\\{F(x)-F^{2}(x)\\}\log\bigg{(}\dfrac{6f(x)\\{F(x)-F^{2}(x)\\}}{3f(x)F^{2}(x)}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{1}6(u-u^{2})\log\bigg{(}\dfrac{6(u-u^{2})}{3u^{2}}\bigg{)}du~{}~{}\big{(}\mbox{using}~{}u=F(x)\big{)}$
$\displaystyle=$ $\displaystyle KL(f_{U_{2:3}},f_{W}).$

Thus, the proof of the theorem gets completed. ∎

######  Theorem 2.4.

Suppose $X_{1}$, $X_{2}$, and $X_{3}$ are i.i.d. as $X$ with PDF $f(\cdot)$.
Then,

  * (a)

$KL(f_{X_{CT}},f_{\max})=KL(f_{U_{CT}},f_{W}),$

  * (b)

$KL(f_{\max},f_{X_{CT}})=KL(f_{W},f_{U_{CT}}),$

  * (c)

$KL(f_{\max},f)=KL(f_{W},f_{U}),$

where $f_{U_{CT}}(\cdot)$ and $f_{W}(\cdot)$ as given in Theorems 2.2 and
Theorem 2.3, respectively.

###### Proof.

$(a)$ We have

$\displaystyle KL(f_{X_{CT}},f_{\max})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f_{X_{CT}}(x)\log\bigg{(}\dfrac{f_{X_{CT}}(x)}{f_{\max}(x)}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f(x)\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\big{\\}}$
$\displaystyle\times\log\bigg{(}\dfrac{f_{X_{CT}}(x)}{3f(x)F^{2}(x)}\bigg{)}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+3(1-\lambda_{2})u^{2}\big{\\}}$
$\displaystyle\times\log\bigg{(}\dfrac{\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+3(1-\lambda_{2})u^{2}\big{\\}}}{3u^{2}}\bigg{)}du~{}~{}\big{(}\mbox{using}~{}u=F(x)\big{)}$
$\displaystyle=$ $\displaystyle KL(f_{U_{CT}},f_{W}).$

The proofs for $(b)$ and $(c)$ are similar, and are therefore omitted. ∎

Next, we present closed-form expressions for the KL divergences between
$f_{U_{CT}}(\cdot)$ with $f_{U}(\cdot)$, $f_{W}(\cdot)$, and
$f_{U_{2:3}}(\cdot)$. The detailed derivations are omitted for the sake of
brevity.

$\displaystyle(a)~{}KL(f_{U},f_{U_{CT}})$ $\displaystyle=$ $\displaystyle
2-\varphi(p)+\varphi(q)-\log(3-\lambda_{1}-\lambda_{2}),$
$\displaystyle(b)~{}KL(f_{U_{CT}},f_{U})$ $\displaystyle=$
$\displaystyle\log(3-\lambda_{1}-\lambda_{2})-\phi(p)+\phi(q)-\frac{2}{3}(3+\lambda_{1}-4\lambda_{2})$
$\displaystyle+\frac{2}{9(1-\lambda_{2})}\big{(}\lambda^{2}_{1}+13\lambda^{2}_{2}-2\lambda_{1}\lambda_{2}-18\lambda_{2}+6\big{)},$
$\displaystyle(c)~{}KL(f_{U_{2:3}},f_{U_{CT}})$ $\displaystyle=$
$\displaystyle\log\Big{(}\dfrac{6}{3-\lambda_{1}-\lambda_{2}}\Big{)}-\frac{5}{3}-\eta(p)+\eta(q)$
$\displaystyle-\frac{1}{9(1-\lambda_{2})^{2}}\big{(}55\lambda^{2}_{2}-8\lambda^{2}_{1}-44\lambda_{1}\lambda_{2}-78\lambda_{2}+60\lambda_{1}+15\big{)},$
$\displaystyle(d)~{}KL(f_{U_{CT}},f_{W})$ $\displaystyle=$
$\displaystyle-\phi(p)+\phi(q)-\log\Big{(}\frac{3}{3-\lambda_{1}-\lambda_{2}}\Big{)}+\frac{1}{3}(\lambda_{1}+9\lambda_{2}-4)$
$\displaystyle+\frac{2}{9(1-\lambda_{2})}(\lambda^{2}_{1}+13\lambda^{2}_{2}-2\lambda_{1}\lambda_{2}-18\lambda_{2}+6),$
$\displaystyle(e)~{}KL(f_{W},f_{U_{CT}})$ $\displaystyle=$
$\displaystyle\gamma(q)-\gamma(p)+\log\Big{(}\frac{3}{3-\lambda_{1}-\lambda_{2}}\Big{)}+\frac{(\lambda_{2}-\lambda_{1})}{9(1-\lambda_{2})^{2}}(3-7\lambda_{2}+4\lambda_{1})$
$\displaystyle+\frac{2}{9(1-\lambda_{2})^{2}}(4\lambda^{2}_{1}+10\lambda^{2}_{2}-8\lambda_{1}\lambda_{2}-9\lambda_{2}+3)-\frac{2}{3},$
$\displaystyle(f)~{}KL(f_{U_{2:3}},f_{W})$ $\displaystyle=$
$\displaystyle\log(2),$ $\displaystyle(g)~{}KL(f_{W},f_{U})$ $\displaystyle=$
$\displaystyle\log(3)-\frac{2}{3},$

where $p$, $q$ and $\phi(\cdot)$ are as given in Remark 2.2, and

$\displaystyle\varphi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{r}\\{(\lambda_{2}-\lambda_{1})x+3(1-\lambda_{2})x^{2}\\}\log\Big{(}\frac{x}{x-1}\Big{)},$
$\displaystyle\eta(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\\{6(\lambda_{2}-\lambda_{1})x^{2}-(18+4\lambda_{1}-22\lambda_{2})x^{3}+12(1-\lambda_{2})x^{4}\\}\log\Big{(}\frac{x}{x-1}\Big{)},$
$\displaystyle\gamma(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\\{2(\lambda_{2}-\lambda_{1})x^{3}+6(1-\lambda_{2})x^{4}\\}\log\Big{(}\frac{x}{x-1}\Big{)}.$

From the above expressions, it is clear that the KL divergences
$KL(f_{X_{CT}},f)$, $KL(f,f_{X_{CT}})$, $KL(f_{2:3},f_{X_{CT}})$,
$KL(f_{2:3},f_{\max})$, $KL(f_{X_{CT}},f_{\max})$, $KL(f_{\max},f_{X_{CT}})$,
$KL(f_{\max},f)$ are all free from the underlying parent distribution with CDF
$F(\cdot)$.

Recall that the KL divergence is non-symmetric. However, one may get symmetric
divergence based on KL divergence, known as Jeffreys’ divergence. The
Jeffreys’ divergence between two density functions $f_{1}(\cdot)$ and
$f_{2}(\cdot)$ is defined as (see Jeffreys (1946))

$\displaystyle
J_{d}(f_{1},f_{2})=\int_{-\infty}^{\infty}\\{f_{1}(x)-f_{2}(x)\\}\log\Big{(}\frac{f_{1}(x)}{f_{2}(x)}\Big{)}dx=KL(f_{1},f_{2})+KL(f_{2},f_{1}).$
(2.7)

Note that the Kullback-Leibler divergence is not symmetric, but Jeffreys’
divergence is symmetric, that is,

$\displaystyle J_{d}(f_{1},f_{2})=J_{d}(f_{2},f_{1}).$

In the following remark, it is shown that Jeffreys’ divergences between
$f_{X_{CT}}(\cdot)$, $f_{\max}(\cdot)$, and $f_{X_{CT}}(\cdot)$, $f(.)$ are
equal to that between $f_{U_{CT}}(\cdot)$, $f_{W}(\cdot)$, and
$f_{U_{CT}}(\cdot)$, $f_{U}(\cdot)$, respectively. This statement can be
proved using the results presented in Theorem 2.2 and Theorem 2.4.

######  Remark 2.3.

We have

$\displaystyle J_{d}(f_{X_{CT}},f_{\max})$ $\displaystyle=$ $\displaystyle
J_{d}(f_{U_{CT}},f_{W})=KL(f_{W},f_{U_{CT}})+KL(f_{U_{CT}},f_{W}),$
$\displaystyle J_{d}(f_{X_{CT}},f)$ $\displaystyle=$ $\displaystyle
J_{d}(f_{U_{CT}},f_{U})=KL(f_{U},f_{U_{CT}})+KL(f_{U_{CT}},f_{U}).$

Note that like Kullback-Leibler divergence, here, Jeffreys’ divergence is also
free from the underlying distribution $F(\cdot).$

Now, we obtain Kullback-Leibler divergence between a mixture distribution and
CT distribution in the following example.

######  Example 2.3.

Suppose the PDF of a general mixture distribution is
$f_{\text{mix}}(x)=vf(x)+3(1-v)f(x)F^{2}(x)$, $v\in[0,1]$. The Kullback-
Leibler divergence between $f_{mix}(\cdot)$ and $f_{X_{CT}}(\cdot)$ and
$f_{X_{CT}}(\cdot)$ and $f_{mix}(\cdot)$ are, respectively, obtained as

$\displaystyle(a)~{}KL(f_{mix},f_{X_{CT}})$ $\displaystyle=$
$\displaystyle\log\Big{(}\frac{3-2v}{3-\lambda_{1}-\lambda_{2}}\Big{)}+\frac{(1-v)}{9(1-\lambda_{2})^{2}}(4\lambda^{2}_{1}+13\lambda^{2}_{2}-5\lambda_{1}\lambda_{2}-3\lambda_{1}$
$\displaystyle-15\lambda_{2}+6)+\frac{4v^{\frac{3}{2}}}{3\sqrt{3}\sqrt{1-v}}\tan^{-1}\bigg{(}\sqrt{\frac{3(1-v)}{v}}\bigg{)}-\frac{2}{3}(1-2v)$
$\displaystyle-\xi(p)+\xi(q),$ $\displaystyle(b)~{}KL(f_{X_{CT}},f_{mix})$
$\displaystyle=$
$\displaystyle\log\Big{(}\frac{3-\lambda_{1}-\lambda_{2}}{3-2v}\Big{)}-\frac{1}{3}(4+5\lambda_{1}-9\lambda_{2})-\phi(p)+\phi(q)$
$\displaystyle-\frac{v}{3(1-v)}\bigg{\\{}2(1-\lambda_{2})-(\lambda_{2}-\lambda_{1})\log\Big{(}\frac{v}{3-2v}\Big{)}\bigg{\\}}$
$\displaystyle-2\bigg{\\{}\lambda_{1}-\frac{v(1-\lambda_{2})}{3(1-v)}\bigg{\\}}\sqrt{\frac{v}{3(1-v)}}\tan^{-1}\bigg{(}\sqrt{\frac{3(1-v)}{v}}\bigg{)}$
$\displaystyle+\frac{2}{9(1-\lambda_{2})}\big{(}\lambda^{2}_{1}+13\lambda^{2}_{2}-2\lambda_{1}\lambda_{2}-18\lambda_{2}+6\big{)},$

where
$\xi(x)$=$\frac{1}{2r}\big{\\{}2v(\lambda_{2}-\lambda_{1})x+6v(1-\lambda_{2})x^{2}+2(\lambda_{2}-\lambda_{1})(1-v)x^{3}+6(1-v)(1-\lambda_{2})x^{4}\big{\\}}\log\Big{(}\frac{x}{x-1}\Big{)}$,
$r=\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{1}\lambda_{2}-3\lambda_{1}},$
and the expression of $\phi(x)$ is as provided in Remark 2.2.

Using the values of $KL(f_{mix},f_{X_{CT}})$ and $KL(f_{X_{CT}},f_{mix})$, the
Jeffreys’ divergence between a mixture distribution and CT distribution can be
easily computed.

###  2.3 Cubic transmuted Shannon entropy

We now propose an extension of the Shannon entropy of CT distribution as well
as JS divergence, and call it as CT Shannon entropy. The definition of the CT
Shannon entropy is given below. Here, we need the PDF of $X_{CT}$, which is
given by

$\displaystyle
f_{X_{CT}}(x)=\lambda_{1}f(x)+\frac{1}{3}(\lambda_{2}-\lambda_{1})f_{2:3}(x)+\frac{1}{3}(3-\lambda_{2}-2\lambda_{1})f_{\max}(x),$
(2.8)

where $\lambda_{1}\in[0,1]$ and $\lambda_{2}\in[-1,1)$. The CDF of $X_{CT}$ is
denoted by $F_{X_{CT}}(x)$, which can be easily obtained from (2.8). Equation
(2.8) can be obtained from (1.11) by using $F(\cdot),$ $F_{2:3}(\cdot)$ and
$F_{\max}(\cdot)$.

######  Definition 2.1.

Let $X$ be a random variable with PDF $f(\cdot)$ and $X_{CT}$ be a cubic
transmuted random variable with PDF $f_{X_{CT}}(\cdot)$. Then, the CT Shannon
entropy between the PDF $f_{X_{CT}}(\cdot)$ in (2.8) and the PDFs of its
components $f(\cdot)$, $f_{2:3}(\cdot)$ and $f_{\max}(\cdot)$ is defined as

$CTS(f,f_{2:3},f_{\max};\Lambda)=H(f_{X_{CT}})-\lambda_{1}H(f)-\frac{1}{3}(\lambda_{2}-\lambda_{1})H(f_{2:3})-\frac{1}{3}(3-\lambda_{2}-2\lambda_{1})H(f_{\max}),$
(2.9)

where $\Lambda=(\lambda_{1},\lambda_{2})$, $\lambda_{1}\in[0,1],$ and
$\lambda_{2}\in[-1,1)$.

Now, we establish an important theorem regarding CT Shannon entropy which
shows that $CTS(f,f_{2:3},f_{\max};\Lambda)$, defined in Definition 2.1, can
be expressed in terms of the Kullback-Leibler divergence measures between
$f_{\max}(\cdot)$ and $f(\cdot),$ $f_{2:3}(\cdot)$ and $f_{\max}(\cdot)$, and
$f_{X_{CT}}(\cdot)$ and $f(\cdot)$.

######  Theorem 2.5.

The CT Shannon entropy can be expressed as

$\displaystyle CTS(f,f_{2:3},f_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle(1-\lambda_{2})KL(f_{\max},f)+\frac{1}{3}(\lambda_{2}-\lambda_{1})KL(f_{2:3},f_{\max})$
(2.10) $\displaystyle-KL(f_{X_{CT}},f)+C^{*}$

where $\Lambda=(\lambda_{1},\lambda_{2})$ and
$C^{*}=\big{\\{}\log(3)-1\big{\\}}(\lambda_{2}-\lambda_{1})$.

###### Proof.

Substituting $H(f_{X_{CT}})$ from Theorem 2.1 into (2.9), we obtain

$\displaystyle CTS(f,f_{2:3},f_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle\frac{2}{3}(\lambda_{1}-\lambda_{2})H(f_{\max})-\frac{1}{3}(\lambda_{2}-\lambda_{1})H(f_{2:3})+H(f_{U_{CT}})$
(2.11)
$\displaystyle-(1-\lambda_{2})H(f_{W})+2(\lambda_{2}-\lambda_{1})H^{F}(f)$
$\displaystyle=$
$\displaystyle\frac{1}{3}(\lambda_{1}-\lambda_{2})\left(2H(f_{\max})+H(f_{2:3})-6H^{F}(f)\right)+H(f_{U_{CT}})$
$\displaystyle-(1-\lambda_{2})H(f_{W})$ $\displaystyle=$
$\displaystyle-\frac{1}{3}(\lambda_{1}-\lambda_{2})\big{\\{}KL(f_{2:3},f_{\max})+3(\log
3-1)\big{\\}}+H(f_{U_{CT}})$ $\displaystyle-(1-\lambda_{2})H(f_{W}).$

Now, using the following relations (see Theorem 2.2 and Theorem 2.4)

$KL(f_{X_{CT}},f)=KL(f_{U_{CT}},f_{U})=-H(f_{U_{CT}})~{}\mbox{and}~{}KL(f_{\max},f)=KL(f_{W},f_{U})=-H(f_{W})$

in (2.11), the desired result follows. Hence, the theorem. ∎

Further, bounds of CT Shannon entropy can be found easily in terms of the
Kullback-Leibler divergence measures between various components, as follows:

$\displaystyle
CTS(f,f_{2:3},f_{\max};\Lambda)\left\\{\begin{array}[]{ll}\geq(1-\lambda_{2})KL(f_{\max},f)+\frac{1}{3}(\lambda_{2}-\lambda_{1})KL(f_{2:3},f_{\max})\\\
-KL(f_{X_{CT}},f),~{}~{}~{}\mbox{if}~{}\lambda_{2}\geq\lambda_{1},\\\
\leq(1-\lambda_{2})KL(f_{\max},f)+\frac{1}{3}(\lambda_{2}-\lambda_{1})KL(f_{2:3},f_{\max})\\\
-KL(f_{X_{CT}},f),~{}~{}~{}\mbox{if}~{}\lambda_{2}<\lambda_{1}.\end{array}\right.$
(2.16)

A closed-form expression of $CTS(f,f_{2:3},f_{max};\Lambda)$ can be easily
obtained since

$\displaystyle CTS(f,f_{2:3},f_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle(1-\lambda_{2})KL(f_{W},f_{U})+\frac{1}{3}(\lambda_{2}-\lambda_{1})KL(f_{U_{2:3}},f_{W})$
(2.17) $\displaystyle-
KL(f_{U_{CT}},f_{U})-\big{\\{}\log(3)-1\big{\\}}(\lambda_{1}-\lambda_{2}),$

where $KL(f_{W},f_{U})$, $KL(f_{U_{2:3}},f_{W})$ and $KL(f_{U_{CT}},f_{U})$
are as given above. From (2.17), it is not hard to notice that the CT Shannon
entropy is also free from the underlying distribution.

######  Remark 2.4.

We observe that the Jensen-Shannon (JS) entropy for $f(\cdot)$,
$f_{2:3}(\cdot)$, and $f_{\max}(\cdot)$ is a special case of CT Shannon
entropy when $\lambda_{2}\geq\lambda_{1}$ and
$\lambda_{1},\lambda_{2}\in[0,1).$ Denote $\delta_{1}=\lambda_{1}$,
$\delta_{2}=\frac{1}{3}(\lambda_{2}-\lambda_{1})$ and
$\delta_{3}=\frac{1}{3}(3-\lambda_{2}-2\lambda_{1})$. Then, evidently,

$CTS(f,f_{2:3},f_{\max};\lambda_{1},\lambda_{2})=JS(f,f_{2:3},f_{\max};\delta),$

where
$JS(f,f_{2:3},f_{\max};\delta)=H(\delta_{1}f+\delta_{2}f_{2:3}+\delta_{3}f_{\max})-\\{\delta_{1}H(f)+\delta_{2}H(f_{2:3})+\delta_{3}H(f_{\max})\\}$
with $\delta=(\delta_{1},\delta_{2},\delta_{3})$.

######  Remark 2.5.

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1-\lambda$, (2.10) reduces to

$\displaystyle CTS(f,f_{2:3},f_{\max};\lambda)$ $\displaystyle=$
$\displaystyle\lambda KL(f_{\max},f)-\frac{2}{3}\lambda
KL(f_{2:3},f_{\max})-KL(f^{*}_{X_{CT}},f)$
$\displaystyle+2\lambda\big{\\{}1-\log(3)\big{\\}},~{}|\lambda|\leq 1,$

which is the CT Shannon entropy of the one-parameter CT distribution.

##  3 Gini information

In this section, we discuss another important information measure, namely,
Gini’s mean difference of CT distribution, CT Gini’s mean difference and
investigate their connection to energy distances.

###  3.1 Gini’s mean difference

Let $X$ be an absolutely continuous random variable with CDF $F(\cdot)$. Then,
the Gini’s mean difference $(GMD)$ of $X$ is given by

$\displaystyle GMD(F)=2\int_{-\infty}^{\infty}F(x)\bar{F}(x)dx,$ (3.1)

where $\bar{F}(x)=1-F(x)$ is the survival function of $X$. We now show that
Gini’s mean difference for the CT distribution can be written in terms of GMDs
of its components.

######  Theorem 3.1.

Let $X$ and $X_{CT}$ be a continuous random variable and its CT random
variable, respectively. Then, the Gini’s mean difference of
$F_{X_{CT}}(\cdot)$ in (1.11) is given by

$\displaystyle GMD(F_{X_{CT}})$ $\displaystyle=$
$\displaystyle\lambda^{2}_{1}GMD(F)+(1-\lambda_{2})^{2}GMD(F_{\max})$ (3.2)
$\displaystyle+2\int_{0}^{1}\bigg{\\{}\dfrac{Au+Bu^{2}+Cu^{3}+Du^{4}+Eu^{5}}{f(F^{-1}(u))}\bigg{\\}}du,$

where $A=\lambda_{1}(1-\lambda_{1})$, $B=\lambda_{2}-\lambda_{1}$,
$C=\lambda_{2}(1-\lambda_{2})-2\lambda_{1}(\lambda_{2}-\lambda_{1})$,
$D=-\big{\\{}(\lambda_{2}-\lambda_{1})^{2}+2\lambda_{1}(1-\lambda_{2})\big{\\}}$
and $E=-2(1-\lambda_{2})(\lambda_{2}-\lambda_{1})$.

###### Proof.

From (3.1), we have

$\displaystyle\frac{1}{2}GMD(F_{X_{CT}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}F_{X_{CT}}(x)\bar{F}_{X_{CT}}(x)dx$
$\displaystyle=$
$\displaystyle\lambda^{2}_{1}\int_{-\infty}^{\infty}F(x)\bar{F}(x)dx+(1-\lambda_{2})^{2}\int_{-\infty}^{\infty}F^{3}(x)\bar{F}^{3}(x)dx$
$\displaystyle+\int_{-\infty}^{\infty}\big{\\{}AF(x)+BF^{2}(x)+CF^{3}(x)+DF^{4}+EF^{5}(x)\big{\\}}dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{2}_{1}GMD(F)+\frac{1}{2}(1-\lambda_{2})^{2}GMD(F_{\max})$
$\displaystyle+\int_{0}^{1}\bigg{\\{}\dfrac{Au+Bu^{2}+Cu^{3}+Du^{4}+Eu^{5}}{f(F^{-1}(u))}\bigg{\\}}du,~{}~{}\big{(}using~{}u=F(x)\big{)}$

which completes the proof of the theorem. ∎

In the following, we present upper and lower bounds for the GMD as

$\displaystyle
GMD(F_{X_{CT}})\left\\{\begin{array}[]{ll}\geq\lambda^{2}_{1}GMD(F)+(1-\lambda_{2})^{2}GMD(F_{\max}),&\mbox{if}~{}R^{*}\geq
0\\\
\leq\lambda^{2}_{1}GMD(F)+(1-\lambda_{2})^{2}GMD(F_{\max}),&\mbox{if}~{}R^{*}\leq
0,\end{array}\right.$ (3.5)

where
$R^{*}=2\int_{0}^{1}\\{(Au+Bu^{2}+Cu^{3}+Du^{4}+Eu^{5})/f(F^{-1}(u))\\}du$,
with $A$, $B$, $C$, $D$, and $E$ being as in Theorem 3.1.

Substituting $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1$ in $R^{*}$, we obtain

$\displaystyle\hat{R}^{*}$ $\displaystyle=$ $\displaystyle
2\lambda^{2}\int_{-\infty}^{\infty}F^{2}(x)\bar{F}^{2}(x)dx-\lambda(1+\lambda)\int_{0}^{1}\frac{u(1-u)(1+2u)}{f(F^{-1}(u))}du.$
(3.6)

######  Remark 3.1.

  * •

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1$ and using (3.6) in (3.2),
Gini’s mean difference of CT distribution becomes

$\displaystyle GMD(F_{X_{T}})$ $\displaystyle=$
$\displaystyle(1+\lambda)^{2}GMD(F)+\lambda^{2}GMD(F_{\max})$
$\displaystyle-\lambda(1+\lambda)\int_{0}^{1}\frac{u(1-u)(1+2u)}{f(F^{-1}(u))}du,$

which is the Gini’s mean difference of transmuted distribution. For the
details, see Kharazmi and Balakrishnan (2021);

  * •

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1-\lambda$, from (3.2), we have

$\displaystyle GMD(F^{*}_{X_{CT}})$ $\displaystyle=$
$\displaystyle(1+\lambda)^{2}GMD(F)+\lambda^{2}GMD(F_{\max})$
$\displaystyle-2\lambda\int_{0}^{1}\bigg{\\{}\dfrac{(1+\lambda)u+2u^{2}-(5+3\lambda^{2})u^{3}+(6\lambda+2)u^{4}-4\lambda
u^{5}}{f(F^{-1}(u))}\bigg{\\}}du,$

which is the Gini’s mean difference of the one-parameter CT distribution.

Next, we present an example to demonstrate the computation of the GMD of a CT
distributed random variable with power distribution as the baseline
distribution.

######  Example 3.1.

Let $X$ be a random variable following power distribution with $CDF$
$F(x)=(\frac{x}{b})^{c},~{}\mbox{for}~{}0<x<b,~{}c>0$. Then, the GMD of the
associated CT distribution is given by

$\displaystyle GMD(F_{X_{TC}})$ $\displaystyle=$
$\displaystyle\frac{2bc}{(c+1)(2c+1)}\lambda^{2}_{1}+2b(1-\lambda_{2})^{2}\bigg{(}\frac{1}{3c+1}-\frac{3}{4c+1}+\frac{3}{5c+1}$
$\displaystyle-\frac{1}{6c+1}\bigg{)}+2b\bigg{(}\frac{A}{c+1}+\frac{B}{2c+1}+\frac{C}{3c+1}+\frac{D}{4c+1}+\frac{E}{5c+1}\bigg{)}.$

where $A$, $B$, $C$, $D$, and $E$ are as given in Theorem 3.1. For $b=2$ and
$c=3$, the term
$R^{*}=2b\bigg{(}\frac{A}{c+1}+\frac{B}{2c+1}+\frac{C}{3c+1}+\frac{D}{4c+1}+\frac{E}{5c+1}\bigg{)}$
has been plotted in Figure 2 and we notice from it that the term $R^{*}$ takes
positive as well as negative values for some $(\lambda_{1},\lambda_{2})$.
Moreover,

$\displaystyle R^{*}(\lambda_{1},\lambda_{2})\left\\{\begin{array}[]{ll}\geq
0,&\mbox{if}~{}\lambda_{1}\in[0,0.8]~{}~{}and~{}~{}\lambda_{2}\in[0.2,1]\\\
\leq
0,&\mbox{if}~{}\lambda_{1}\in[0.8,1]~{}~{}and~{}~{}\lambda_{2}\in[-1,0.2].\end{array}\right.$
(3.9)

Figure 2: Plots of $R^{*}(\lambda_{1},\lambda_{2})$ in Example 3.1.

###  3.2 Cubic Transmuted Gini’s mean difference

In this subsection, we propose an extension of Gini’s mean difference for the
cubic transmuted distribution as well as Jensen-Gini divergence. This extended
version of the measure is called cubic transmuted Gini’s mean difference and
is as defined below.

######  Definition 3.1.

Suppose $X$ and $X_{CT}$ have CDFs $F(\cdot)$ and $F_{X_{CT}}(\cdot)$,
respectively. Then, the cubic transmuted Gini’s mean difference $(CTG)$
between $F_{X_{CT}}(\cdot)$ and its components’ CDFs $F(\cdot)$,
$F_{2:3}(\cdot)$, and $F_{\max}(\cdot)$ is given by

$\displaystyle CTG(F,F_{2:3},F_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle
GMD(F_{X_{CT}})-\lambda_{1}GMD(F)-\frac{1}{3}(\lambda_{2}-\lambda_{1})GMD(F_{2:3})$
(3.10) $\displaystyle-\frac{1}{3}(3-\lambda_{2}-2\lambda_{1})GMD(F_{\max}),$

where $\Lambda=(\lambda_{1},\lambda_{2})$, $\lambda_{1}\in[0,1],$ and
$\lambda_{2}\in[-1,1)$.

We now derive a connection of CT Gini’s mean difference with energy distance
in the following Theorem. Let $X_{1}$ and $X_{2}$ be two random variables with
CDFs $F_{1}(\cdot)$ and $F_{2}(\cdot),$ respectively. Then, the energy
distance between $F_{1}(\cdot)$ and $F_{2}(\cdot)$ is given by

$\displaystyle
CD(F_{1},F_{2})=\int_{-\infty}^{\infty}\big{\\{}F_{1}(x)-F_{2}(x)\big{\\}}^{2}dx.$
(3.11)

######  Theorem 3.2.

The CT Gini’s mean difference can be presented through energy distances
between $F_{X_{CT}}(\cdot)$ with $F(\cdot)$, $F_{2:3}(\cdot)$, and
$F_{\max}(\cdot)$ as follows:

$\displaystyle CTG(F,F_{2:3},F_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle\eta_{1}~{}CD(F,F_{X_{CT}})+\eta_{2}~{}CD(F_{\max},F_{X_{CT}})$
$\displaystyle+(1-\eta_{1}-\eta_{2})~{}CD(F_{2:3},F_{X_{CT}}),$

where

$\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle(\lambda_{1},\lambda_{2}),~{}~{}\eta_{1}=\frac{1}{1-2\lambda_{1}}(4\lambda_{1}-2\lambda_{2}-3\lambda^{2}_{1}),$
$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{9(2\lambda_{2}-3)}\big{\\{}(26\lambda_{2}-20\lambda_{1}-6\lambda^{2}_{2})-3\eta_{1}(4\lambda_{2}-8)\big{\\}}.$

###### Proof.

Using (3.2) in (3.10), we have

$\displaystyle CTG(F,F_{2:3},F_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle(\lambda^{2}_{1}-\lambda_{1})GMD(F(x))-\frac{1}{3}(3\lambda^{2}_{2}-5\lambda_{2}+2\lambda_{1})GMD(F_{\max}(x))$
$\displaystyle+2\int_{-\infty}^{\infty}\big{\\{}AF(x)+BF^{2}(x)+CF^{3}(x)+DF^{4}(x)+EF^{5}(x)\big{\\}}dx$
$\displaystyle-\frac{1}{3}(\lambda_{2}-\lambda_{1})GMD(F_{2:3}(x)).$

Further, using (3.1) in the above equation and after some algebraic
calculations, we obtain

$\displaystyle CTG(F,F_{2:3},F_{\max};\Lambda)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\big{\\{}(4\lambda_{1}-2\lambda^{2}_{1}-2\lambda)F^{2}(x)+\frac{4}{3}(3\lambda^{2}_{1}-3\lambda_{1}\lambda_{2}+\lambda_{2}-\lambda_{1})F^{3}(x)$
$\displaystyle+(12\lambda_{2}-16\lambda_{1}+8\lambda_{1}\lambda_{2}-2\lambda^{2}_{1}-2\lambda^{2}_{2})F^{4}(x)$
$\displaystyle+4(\lambda^{2}_{2}-\lambda_{2}\lambda_{1}+5\lambda_{1}-5\lambda_{2})F^{5}(x)+\frac{2}{3}(\lambda_{2}-10\lambda_{1}-3\lambda^{2}_{2})F^{6}(x)\big{\\}}dx$
$\displaystyle=$
$\displaystyle\eta_{1}~{}CD(F,F_{X_{CT}})+\eta_{2}~{}CD(F_{\max},F_{X_{CT}})$
$\displaystyle+(1-\eta_{1}-\eta_{2})~{}CD(F_{2:3},F_{X_{CT}}),$

where the last equality is obtained using (3.11). Thus, the proof gets
completed. ∎

In the following remark, we present CT Gini’s mean difference for some special
case of the CT distributions.

######  Remark 3.2.

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1-\lambda$, from Theorem 3.2, we
get

$\displaystyle CTG(F,F_{2:3},F_{\max};\lambda)$ $\displaystyle=$
$\displaystyle\eta^{*}_{1}CD(F,F^{*}_{X_{CT}})+\eta^{*}_{2}CD(F_{\max},F^{*}_{X_{CT}})$
$\displaystyle+(1-\eta^{*}_{1}-\eta^{*}_{2})CD(F_{2:3},F^{*}_{X_{CT}})$

as the CT Gini’s mean difference of the one parameter CT distribution, where
$\eta^{*}_{1}=\frac{1}{1+2\lambda}(3\lambda^{2}+2)$ and
$\eta^{*}_{2}=\frac{1}{9(1+2\lambda)}\\{(34\lambda+6\lambda^{2})+12\eta^{*}_{1}(1+\lambda)\\}$.

##  4 Chi-square divergence

In this section, we derive Chi-square $(\chi^{2})$ divergence between the CT
distribution with CDF $F_{X_{CT}}(\cdot)$ given in (1.11) and its components’
CDFs $F(\cdot)$, $F_{2:3}(\cdot)$, and $F_{max}(\cdot)$. Suppose $X$ and $Y$
are two random variables with PDFs $f_{1}(\cdot)$ and $f_{2}(\cdot)$,
respectively. Then, $\chi^{2}$ divergence (see Nielsen and Nock (2013))
between $f_{1}(\cdot)$ and $f_{2}(\cdot)$ is given by

$\displaystyle\chi^{2}(f_{1},f_{2})=\int_{-\infty}^{\infty}\frac{\big{\\{}f_{1}(x)-f_{2}(x)\big{\\}}^{2}}{f_{2}(x)}dx.$
(4.1)

Now, we present some important properties of $\chi^{2}$ divergence in the
following theorem.

######  Theorem 4.1.

Suppose $X_{1}$, $X_{2}$ and $X_{3}$ are i.i.d. as $X$ with PDF $f(\cdot)$.
Then, the Chi-square divergence between $f_{X_{CT}}(\cdot)$ with $f(\cdot)$
and $f_{\max}(\cdot)$ are given by

  * (a)

$\chi^{2}_{CT}(f_{X_{CT}},f)=\chi^{2}_{CT}(f_{U_{CT}},f_{U}),$

  * (b)

$\chi^{2}_{CT}(f,f_{X_{CT}})=\chi^{2}_{CT}(f_{U},f_{U_{CT}}),$

  * (c)

$\chi^{2}_{CT}(f_{\max},f_{X_{CT}})=\chi^{2}_{CT}(f_{W},f_{U_{CT}}).$

###### Proof.

$(a)$ We have

$\displaystyle\chi^{2}_{CT}(f_{X_{CT}},f)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\dfrac{\big{\\{}f_{X_{CT}}(x)-f(x)\big{\\}}^{2}}{f(x)}dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\dfrac{\big{[}f(x)\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\big{\\}}-f(x)\big{]}^{2}}{f(x)}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\big{[}\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})u+3(1-\lambda_{2})u^{2}\big{\\}}-1\big{]}^{2}du~{}~{}\big{(}\mbox{using
u=F(x)}\big{)}$ $\displaystyle=$
$\displaystyle\chi^{2}_{CT}(f_{U_{CT}},f_{U}).$

parts $(b)$ and $(c)$ can be proved similarly. ∎

Closed-form expressions of the Chi-square divergences
$\chi^{2}_{CT}(f_{U_{CT}},f_{U})$, $\chi^{2}_{CT}(f_{U},f_{U_{CT}}),$ and
$\chi^{2}_{CT}(f_{W},f_{U_{CT}})$ are obtained below. The detailed derivations
are omitted here for conciseness:

$\displaystyle(a)~{}\chi^{2}_{CT}(f_{U_{CT}},f_{U})$ $\displaystyle=$
$\displaystyle\frac{1}{15}(12-15\lambda_{1}-9\lambda_{2}+5\lambda_{1}\lambda_{2}+5\lambda^{2}_{1}+2\lambda^{2}_{2}),$
$\displaystyle(b)~{}\chi^{2}_{CT}(f_{U},f_{U_{CT}})$ $\displaystyle=$
$\displaystyle-1+\frac{1}{r}\tanh^{-1}(\lambda),$
$\displaystyle(c)~{}\chi^{2}_{CT}(f_{W},f_{U_{CT}})$ $\displaystyle=$
$\displaystyle-1+\frac{1}{3(1-\lambda_{2})^{3}}(4\lambda^{2}_{1}+10\lambda^{2}_{2}-8\lambda_{1}\lambda_{2}-9\lambda_{2}+3)-9\sigma(p,q),$

where $\lambda=\frac{r}{\lambda_{2}}$ and
$\sigma(p,q)=\frac{1}{2r}\Big{\\{}p^{4}\log\big{(}\frac{p}{p-1}\big{)}-q^{4}\log\big{(}\frac{q}{q-1}\big{)}\Big{\\}}.$

######  Remark 4.1.

Making use of Theorem 4.1 and the expressions of
$\chi^{2}_{CT}(f_{U_{CT}},f_{U})$, $\chi^{2}_{CT}(f_{U},f_{U_{CT}}),$ and
$\chi^{2}_{CT}(f_{W},f_{U_{CT}})$, we observe that Chi-square divergence
measures are free of the underlying distribution function $F(\cdot)$.

The Symmetric $\chi^{2}$-divergence between two density functions
$f_{1}(\cdot)$ and $f_{2}(\cdot)$ is defined as (see Dragomir et al. (2000))

$\displaystyle\Psi(f_{1},f_{2})=\int_{-\infty}^{\infty}\frac{\big{\\{}f_{1}(x)-f_{2}(x)\big{\\}}^{2}\big{\\{}f_{1}(x)+f_{2}(x)\big{\\}}}{f_{1}(x)f_{2}(x)}dx.$
(4.2)

Note that, while the Chi-square divergence is non-symmetric, but (4.2) is
symmetric, that is, $\Psi(f_{1},f_{2})=\Psi(f_{2},f_{1}).$ In the following
remark, we show that the symmetric Chi-square divergence between
$f_{X_{CT}}(\cdot)$ and $f(\cdot)$ is equal to that between
$f_{U_{CT}}(\cdot)$ and $f_{U}(\cdot)$.

######  Remark 4.2.

We have

$\Psi(f_{X_{CT}},f)=\chi^{2}(f_{U_{CT}},f_{U})+\chi^{2}(f_{U},f_{U_{CT}})=\Psi(f_{U_{CT}},f_{U}).$

It is clear that the symmetric Chi-square divergence between
$f_{X_{CT}}(\cdot)$ and $f(\cdot)$ is free from the underlying distribution
$F(\cdot)$.

##  5 Fisher information

In this section, we study Fisher information for the CT distribution with CDF
in (1.11) and for the one-parameter CT distribution with CDF in (1.13).

###  5.1 Fisher information matrix of CT distribution for $\lambda_{1}$ and
$\lambda_{2}$

Below, we present the Fisher information matrix for the cubic transmuted
distribution.

######  Theorem 5.1.

Suppose $X_{CT}$ is a cubic transmuted random variable with PDF
$f_{CT}(\cdot)$ as in (1.12). Then, the Fisher information matrix is

$I(\Lambda)=\begin{bmatrix}E(\tau\rho_{1})&E(\tau\rho_{2})\\\
E(\tau\rho_{2})&E(\tau\rho_{3})\end{bmatrix},$

where $\tau=\frac{f^{2}(x)}{f^{2}_{X_{CT}}}$,
$\rho_{1}\big{\\{}1-2F(x)\big{\\}}^{2}$,
$\rho_{2}=\big{\\{}2F(x)-7F^{2}(x)+6F^{3}(x)\big{\\}}$,
$\rho_{3}=\big{\\{}2F(x)-3F^{2}(x^{2})\big{\\}}^{2}$, and
$\Lambda=(\lambda_{1},\lambda_{2})$.

###### Proof.

Using log-likelihood function

$l(\Lambda)=\log\Big{(}f(x)\big{\\{}\lambda_{1}+2(\lambda_{2}-\lambda_{1})F(x)+3(1-\lambda_{2})F^{2}(x)\big{\\}}\Big{)},$

the Fisher information matrix is easily obtained. ∎

We now present an example to illustrate the computation of the Fisher
information matrix for the CT uniform distribution.

######  Example 5.1.

Let the baseline distribution for the CT distribution be uniform in $(0,1)$.
Then, the Fisher information matrix of the CT uniform distribution is given by

$\displaystyle I(\Lambda)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}E(\tau\rho_{1})&E(\tau\rho_{2})\\\
E(\tau\rho_{2})&E(\tau\rho_{3})\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\frac{1}{r}\tanh^{-1}(\lambda)+\omega_{1}+\dfrac{4}{3(1-\lambda_{2})}&-\omega_{2}-\dfrac{4(1-\lambda_{1})}{3(1-\lambda_{2})^{2}}\\\
-\omega_{2}-\dfrac{4(1-\lambda_{1})}{3(1-\lambda_{2})^{2}}&\frac{1}{3(1-\lambda_{2})^{3}}(4\lambda^{2}_{1}-8\lambda_{1}+3\lambda_{2}+1)-\omega_{3}\end{bmatrix},$

where $r,~{}p,~{}q$ are as given in Remark 2.2, $\lambda=r/\lambda_{2},$ and

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{r}\bigg{\\{}(p-p^{2})\log\Big{(}\frac{p}{p-1}\Big{)}-(q-q^{2})\log\Big{(}\frac{q}{q-1}\Big{)}\bigg{\\}},$
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\bigg{\\{}(2p-7p^{2}+6p^{3})\log\Big{(}\frac{p}{p-1}\Big{)}-(2q-7q^{2}+6q^{3})\log\Big{(}\frac{q}{q-1}\Big{)}\bigg{\\}},$
$\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2r}\bigg{\\{}(4p^{2}-12p^{3}+9p^{4})\log\Big{(}\frac{p}{p-1}\Big{)}-(4q^{2}-12q^{3}+9q^{4})\log\Big{(}\frac{q}{q-1}\Big{)}\bigg{\\}}.$

For $\lambda_{1}=1+\lambda$ and $\lambda_{2}=1-\lambda$, the CT distribution
with CDF $F_{X_{CT}}(\cdot)$ in (1.11) reduces to the one-parameter CT
distribution with CDF $F^{*}_{X_{CT}}(\cdot)$ in (1.13), which discussed by
AL-Kadim and Mohammed (2017). In the next subsection, we discuss Fisher
information for this special case.

###  5.2 Fisher information for one-parameter CT distribution

The Fisher information of a continuous random variable $X$ with PDF
$f(x;\theta)$, where $\theta$ is a parameter, is given by (see Fisher (1929))

$\displaystyle\textit{I}(\theta)=\int_{-\infty}^{\infty}\bigg{[}\frac{\partial\log\big{(}f(x;\theta)\big{)}}{\partial\theta}\bigg{]}^{2}f(x;\theta)dx.$

We now establish that the Fisher information for the one-parameter CT
distribution can be represented in terms of Chi-square divergence.

######  Theorem 5.2.

The Fisher information for the one-parameter CT distribution with PDF
$f^{*}_{X_{TC}}(\cdot)$ in (1.14) is given by

$\displaystyle\textit{I}(\lambda)=\frac{1}{\lambda^{2}}\chi^{2}(f,f^{*}_{X_{TC}}).$

###### Proof.

Using the definition of Fisher information for the parameter $\lambda$, we
have

$\displaystyle\textit{I}(\lambda)$ $\displaystyle=$ $\displaystyle
E\bigg{[}\frac{\partial}{\partial\lambda}\log\big{(}f^{*}_{X_{CT}}(X)\big{)}\bigg{]}^{2}$
$\displaystyle=$ $\displaystyle
E\bigg{[}\frac{f(X)\big{\\{}1-4F(X)+3F^{2}(X)\big{\\}}}{f^{*}_{X_{CT}}(X)}\bigg{]}^{2}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\bigg{[}\frac{f(x)\big{\\{}1-4F(x)+3F^{2}(x)\big{\\}}}{f^{*}_{X_{CT}}(x)}\bigg{]}^{2}f^{*}_{X_{CT}}(x)dx$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}}\int_{-\infty}^{\infty}\dfrac{\big{\\{}f(x)-f^{*}_{X_{CT}}(x)\big{\\}}^{2}}{f^{*}_{X_{CT}}(x)}dx$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}}\chi^{2}(f,f^{*}_{X_{CT}}).$

Hence, the desired result. ∎

Closed-form expression of $\textit{I}(\lambda)$ can be easily obtained as

$\displaystyle\textit{I}(\lambda)=\frac{1}{\lambda^{2}}\chi^{2}(f_{U},f^{*}_{U_{TC}})=\frac{1}{\lambda^{2}}\bigg{\\{}\frac{\lambda^{2}}{r^{*}}\tanh^{-1}\bigg{(}\frac{r^{*}}{1-\lambda}\bigg{)}+\pi(p^{*})-\pi(q^{*})-\frac{\lambda}{3}(13\lambda+10)\bigg{\\}},$

where $r^{*}=\sqrt{\lambda^{2}-3\lambda}$,
$p^{*}=\frac{r^{*}-2\lambda}{3\lambda}$,
$q^{*}=\frac{r^{*}+2\lambda}{3\lambda}$, and
$\pi(x)=\frac{\lambda^{2}}{2r^{*}}(9x^{4}-24x^{3}+22x^{2}-8x)\log\big{(}\frac{x}{x-1}\big{)}$.

######  Remark 5.1.

We observe from Theorem 4.1 that the Fisher information for the one-parameter
CT distribution is free from the underlying distribution $F^{*}(\cdot)$ and it
is non-symmetric, but the Fisher information of transmuted distribution
obtained by Kharazmi and Balakrishnan (2021) is symmetric.

##  6 Simulation study

In this section, we carry out a simulation study to demonstrate the importance
of Kullback-Leibler divergence for the model/component selection. The
simulation has been conducted by using $R$ software, based on $500$
replications. First, we generate cubic transmuted uniform random variables
using the following algorithm.

Algorithm:
—————————————————————————————————————

  * Step-1:

Generate $V$ from $U(0,1)$ distribution;

  * Step-2:

Generate three independent $U(0,1)$ distributed random variables, say
$X_{1},~{}X_{2},$ and $X_{3};$

  * Step-3:

Arrange $X_{1},~{}X_{2},$ and $X_{3}$ in ascending order and denote them by
$X_{1:3}<X_{2:3}<X_{3:3};$

  * Step-4:

If $V<\pi_{1},$ then $X_{CT}=X_{1:3}$; but, if $\pi_{1}<V<\pi_{2},$ then
$X_{CT}=X_{2:3}$; otherwise $X_{CT}=X_{3:3}$;

  * Step-5:

To generate $n$ cubic transmuted uniform random variables, repeat Steps
$1$-$4$ $n$ times.

—————————————————————————————————————

Now, we compute $KL(f_{X_{CT}},f_{1:3})~{}(=KL1)$,
$KL(f_{X_{CT}},f_{2:3})~{}(=KL2)$, and $KL(f_{X_{CT}},f_{3:3})~{}(=KL3)$ from
the simulated data sets for different choices of mixing proportions
$(\pi_{1},\pi_{2},\pi_{3})$ and sample sizes $(n)$. In addition, we also
compute three model/component proportions based on $KL1$, $KL2$, and $KL3$.
All the computed values with difference choices of mixing proportions and
sample sizes are presented in Table 1. From Table 1, we notice the following:

  * (i)

The values of KL divergences (here $KL1$, $KL2$, and $KL3$) maintain the order
corresponding to the values of mixing proportions, as one would expect. For
example, when $\pi_{1}=0.9$, $\pi_{2}=0.05$, and $\pi_{3}=0.05$, the $KL1$
takes smaller value than $KL2$ and $KL3$, which is clearly observed in Table
1;

  * (ii)

From Table 1, we also notice that the model/component proportion is largest
corresponding to the largest mixing proportion. Further, the model/component
proportions get improved when the sample size increases.

Based on the above observations, we can conclude that the Kullback-Leibler
divergence can be quite useful as a model selection criterion.

Table 1: The KL divergences and model/component proportions for different
choices of mixing proportions and sample sizes.

Mixing proportions | $\bf{n}$ | $\bf{KL1}$ | KL2 | KL3 | Proportion (KL1) | Proportion (KL2) | Proportion (KL3)
---|---|---|---|---|---|---|---
$(0.9,0.05,0.05)$ | 150 | 0.1461 | 0.2550 | 0.8350 | 0.954 | 0.046 | 0.000
| 300 | 0.1593 | 0.2548 | 0.8874 | 0.990 | 0.010 | 0.000
| 500 | 0.0671 | 0.2625 | 0.8963 | 1.000 | 0.000 | 0.000
$(0.05,0.9,0.05)$ | 150 | 0.0917 | 0.0554 | 0.1712 | 0.236 | 0.764 | 0.000
| 300 | 0.0886 | 0.0459 | 0.1483 | 0.050 | 0.950 | 0.000
| 500 | 0.3327 | 0.0497 | 0.1598 | 0.000 | 1.000 | 0.000
$(0.05,0.05,0.9)$ | 150 | 0.8201 | 0.2103 | 0.0836 | 0.000 | 0.002 | 0.998
| 300 | 0.829 | 0.3062 | 0.1483 | 0.000 | 0.000 | 1.000
| 500 | 1.341 | 0.3432 | 0.1045 | 0.000 | 0.000 | 1.000
$(0.9,0.075,0.025)$ | 150 | 0.1461 | 0.2550 | 0.8350 | 0.954 | 0.046 | 0.000
| 300 | 0.1593 | 0.2548 | 0.8874 | 0.990 | 0.010 | 0.000
| 500 | 0.0671 | 0.2625 | 0.8963 | 1.000 | 0.000 | 0.000
$(0.075,0.9,0.025)$ | 150 | 0.0758 | 0.0586 | 0.1748 | 0.424 | 0.576 | 0.000
| 300 | 0.0719 | 0.0500 | 0.1532 | 0.242 | 0.758 | 0.000
| 500 | 0.3102 | 0.0537 | 0.1648 | 0.000 | 1.000 | 0.000
$(0.1,0.8,0.1)$ | 250 | 0.0804 | 0.0571 | 0.1009 | 0.214 | 0.746 | 0.040
| 300 | 0.0857 | 0.0599 | 0.0938 | 0.154 | 0.778 | 0.068
| 500 | 0.3427 | 0.0606 | 0.1003 | 0.000 | 0.994 | 0.006
$(0.1,0.1,0.8)$ | 150 | 0.6503 | 0.1225 | 0.1031 | 0.000 | 0.338 | 0.662
| 300 | 0.6451 | 0.197 | 0.1125 | 0.000 | 0.000 | 1.000
| 500 | 1.1142 | 0.2300 | 0.1171 | 0.000 | 0.000 | 1.000
$(0.3,0.6,0.1)$ | 30 | 0.1841 | 0.1628 | 0.2229 | 0.362 | 0.638 | 0.000
| 40 | 0.1440 | 0.0953 | 0.2459 | 0.224 | 0.776 | 0.000
| 50 | 0.1814 | 0.1005 | 0.2623 | 0.030 | 0.970 | 0.000

Further, to obtain $100\times(1-\gamma)\%$ confidence intervals for the
unknown model parameters of the cubic transmuted Weibull (CTW) distribution,
we used the Fisher information matrix, provided in Section $5$. The
probability density function of the CTW distribution is given by

$\displaystyle
f_{X_{CT}}(x)=kx^{k-1}e^{-x^{k}}\\{\lambda_{1}+2(\lambda_{2}-\lambda_{1})(1-e^{-x^{k}})+3(1-\lambda_{2})(1-e^{-x^{k}})^{2}\\},$
(6.1)

where $x>0,~{}k>0,~{}\lambda_{1}\in[0,1]$ and $\lambda_{2}\in[-1,1)$. Using
the inverse Fisher information matrix as the variance-covariance matrix
computed at MLEs of parameters, $100\times(1-\gamma)\%$ approximate confidence
intervals of unknown model parameters were obtained and are provided in Table
2. Here, we obtained $90\%$ and $95\%$ confidence intervals for different
choices of the parameters and sample sizes $(n)$. It is seen from Table 2 that
as $n$ increases, the average width of the intervals decrease. In addition, we
have also considered cubic transmuted uniform distribution with probability
density function, given by

$\displaystyle
f_{X_{CT}}(x)=\lambda_{1}+2(\lambda_{2}-\lambda_{1})x+3(1-\lambda_{2})x^{2},~{}x\in[0,1],~{}\lambda_{1}\in[0,1],~{}\lambda_{2}\in[-1,1),$
(6.2)

and obtained the approximate confidence intervals for the model parameters,
and these are presented in Table 3. From its, we notice similar observations
as in the case of CTW distribution.

Table 2:  $90\%$ and $95\%$ confidence intervals for the model parameters of
CTW distribution.

Parameters | $\bf{n}$ | 90% confidence interval | 95% confidence interval
---|---|---|---
$\lambda_{1}=0.4$ | 150 | $(0.3852,~{}0.5874)$ | $(0.3504,~{}0.6221)$
| 300 | $(0.3738,~{}0.5360)$ | $(0.346,~{}0.5638)$
| 500 | $(0.3318,~{}0.4489)$ | $(0.3117,~{}0.469)$
$\lambda_{2}=0.6$ | 150 | $(0.1245,~{}0.5328)$ | $(0.0544,~{}0.6029)$
| 300 | $(0.2825,~{}0.5885)$ | $(0.2299,~{}0.6410)$
| 500 | $(0.4679,~{}0.6971)$ | $(0.4285,~{}0.7364)$
$k=1.0$ | 150 | $(1.0018,~{}1.0675)$ | $(0.9906,~{}1.0788)$
| 300 | $(0.9900,~{}1.0389)$ | $(0.9816,~{}1.0473)$
| 500 | $(1.0082,~{}1.0465)$ | $(1.0016,~{}1.0531)$
$\lambda_{1}=0.7$ | 150 | $(0.5658,~{}0.7978)$ | $(0.526,~{}0.8376)$
| 300 | $(0.7307,~{}0.928)$ | $(0.6968,~{}0.9618)$
| 500 | $(0.6455,~{}0.7963)$ | $(0.6196,~{}0.8222)$
$\lambda_{2}=0.5$ | 150 | $(0.2113,~{}0.6495)$ | $(0.1361,~{}0.7247)$
| 300 | $(0.0099,~{}0.3560)$ | $(0.0000,~{}0.4154)$
| 500 | $(0.3027,~{}0.5696)$ | $(0.2569,~{}0.6154)$
$k=1.5$ | 150 | $(1.4705,~{}1.5756)$ | $(1.4524,~{}1.5937)$
| 300 | $(1.5111,~{}1.5893)$ | $(1.4977,~{}1.6027)$
| 500 | $(1.5218,~{}1.5854)$ | $(1.5108~{},1.5963)$
$\lambda_{1}=0.8$ | 150 | $(0.5991,~{}0.8646)$ | $(0.5535,~{}0.9102)$
| 300 | $(0.856,~{}1.0741)$ | $(0.8186,~{}1.1116)$
| 500 | $(0.7852,~{}0.964)$ | $(0.7545,~{}0.9947)$
$\lambda_{2}=0.8$ | 150 | $(0.6285,~{}1.1027)$ | $(0.5471,~{}1.1841)$
| 300 | $(0.2188,~{}0.5915)$ | $(0.1548~{},0.6555)$
| 500 | $(0.4928,~{}0.7981)$ | $(0.4404,~{}0.8505)$
$k=2.0$ | 150 | $(1.9274,~{}2.0909)$ | $(1.8993,~{}2.1189)$
| 300 | $(2.0405,~{}2.1589)$ | $(2.0202,~{}2.1792)$
| 500 | $(2.0535,~{}2.1543)$ | $(2.0362~{},2.1716)$

Table 3:  $90\%$ and $95\%$ confidence intervals of the parameters of cubic
transmuted uniform distribution.

Parameters | $\bf{n}$ | 90% confidance interval | 95% confidance interval
---|---|---|---
$\lambda_{1}=0.4$ | 150 | $(0.3483,~{}0.5198)$ | $(0.3188,~{}0.5492)$
| 300 | $(0.3625,~{}0.4931)$ | $(0.34,~{}0.5155)$
| 500 | $(0.2973,~{}0.3909)$ | $(0.2812,~{}0.407)$
$\lambda_{2}=0.6$ | 150 | $(0.2539,~{}0.6138)$ | $(0.1921,~{}0.6756)$
| 300 | $(0.3588,~{}0.6131)$ | $(0.3151,~{}0.6567)$
| 500 | $(0.5729,~{}0.7660)$ | $(0.5398,~{}0.7992)$
$\lambda_{1}=0.7$ | 150 | $(0.5575,~{}0.752)$ | $(0.5241,~{}0.7854)$
| 300 | $(0.6742,~{}0.828)$ | $(0.6478,~{}0.8543)$
| 500 | $(0.5836,~{}0.6978)$ | $(0.5639,~{}0.7174)$
$\lambda_{2}=0.5$ | 150 | $(0.2989,~{}0.6669)$ | $(0.2358,~{}0.7309)$
| 300 | $(0.1891,~{}0.4607)$ | $(0.1424,~{}0.5073)$
| 500 | $(0.4784,~{}0.6845)$ | $(0.443,~{}0.7199)$
$\lambda_{1}=0.8$ | 150 | $(0.6198,~{}0.8259)$ | $(0.5844,~{}0.8613)$
| 300 | $(0.759,~{}0.9192)$ | $(0.7315,~{}0.9467)$
| 500 | $(0.6803,~{}0.8022)$ | $(0.6593,~{}0.8231)$
$\lambda_{2}=0.8$ | 150 | $(0.7016,~{}1.0637)$ | $(0.6394,~{}1.1258)$
| 300 | $(0.4927,~{}0.7660)$ | $(0.4457~{},0.8129)$
| 500 | $(0.7771,~{}0.986)$ | $(0.7412,~{}1.0219)$

##  7 Conclusions

In this paper, we have studied various informational properties of the family
of cubic transmuted distributions. The Shannon entropy and Gini’s mean
difference of the CT distributions are derived. Further, we have proposed CT
Shannon entropy and CT Gini’s mean difference. Connection between CT Shannon
entropy and Kullback-Leibler divergence and that between CT Gini’s mean
difference and energy distance have also been established. Chi-square
divergence between the CT distribution and the PDFs of its components have
been obtained. Fisher information matrix is obtained for the CT distribution,
and we also derive Fisher information for the one-parameter cubic transmuted
distribution. Some simulation results have also been presented to demonstrate
the usefulness of some of the informational measures studied here in the
context of inference.

## Acknowledgments

The authors thank the referees for all their helpful comments and suggestions,
which led to the substantial improvements. Shital Saha thanks the University
Grants Commission (Award No. $191620139416$), India, for financial
assistantship to carry out this research work. The first two authors thank the
research facilities provided by the Department of Mathematics, National
Institute of Technology Rourkela, India.

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