# Characterizations and Entropy Measures of the Exponentiated Generalized Frechet Geometric Distribution

Authors: Naz Saud; Aqsa Rafique; Muhammad Ijaz; Naila Amjad; Mahmoud El-Morshedy; Syed Habib Shah
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Some characterizations and entropy measures of the Exponentiated Generalized Fréchet Geometric (EGFG) distribution are studied in this paper. Firstly, characterizations of the EGFG distribution based on five different approaches are discussed. The submodels for the EGFG distribution with their characterization expressions formed on the ratio of two truncated moments are also presented. Secondly, four different entropy measures are considered and expressed analytically via the incomplete gamma function. The behavior of all these entropy measures is discussed by performing a numerical study.


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## Body

## 1. Introduction

Characterization is a distributional property of statistics that determines the associated stochastic model. In designing a stochastic model for a particular modeling problem, it is crucial to validate whether the specified probability distribution fulfills the underlying conditions by its characterization before a specified probability model is used to fit data applications. Thus, it will rely on the characterization of the chosen distribution. It plays an essential role in different fields of mathematical and statistical sciences.Although in several statistical applications, an increase in the number of parameters gives a more appropriate model, and a smaller number of parameters (without affecting the model suitability) is mathematically more attractive in characterization complications. Over the years, many researchers have studied the different techniques of characterization of continuous probability distributions such as Glänzel [1]; Glänzel [2]; Hamedani [3]; Hamedani [4]; Hamedani [5]; Hamedani and Ahsanullah [6]; Hamedani [7]; Bhatti et al. [8]; and Rafique and Saud [9]. Consequently, numerous characterization techniques have been studied in this paper.Entropy is used to measure the amount of uncertainty, randomness, and disorderness in the system. These essential mathematical techniques are used to evaluate the uncertainty of the stochastic variable. To be more specific, the distribution of a random variable is related to some level of uncertainty, which is quantified by entropy.In any stochastic process, the probability distribution varies over time; hence, it is clear that the entropy or uncertainty of a probability distribution changes over time as well. It becomes knowing how the uncertainty changes over time. The application of entropy measures is useful in many branches of science and disciplines. Several examples from different areas where different entropy concepts were used, e.g., topological entropy, Kolmogorov–Sinai entropy, topological order, can be found in the following references proposed by Caneco et al. [10], Rocha and Aleixo [11], and Rocha and Carvalho [12].The essential references in this regard are briefly presented below. Shannon [13] introduced the concept of entropy. Cho et al. [14] estimated the entropy of a Weibull distribution under generalized progressive hybrid censoring. The comparison of different entropy measures was discussed by Dey et al. [15]. Basit et al. [16] studied the entropy measures for weighted and truncated weighted exponential distributions. The entropy measures for the uncertainty quantification of stochastic processes were investigated by Namdari and Li [17], while Ijaz et al. [18] compared different entropy measures for Lomax distribution using relative loss approach.This present work deals with the characterization and entropy measures of the univariate continuous distribution named as “Exponentiated Generalized Fréchet Geometric (EGFG) distribution.” The EGFG distribution was proposed by Rafique and Saud [19]. They studied the mathematical properties of the suggested model and also provided its application study.The probability density function (pdf) and cumulative distribution function (cdf) of the Exponentiated Generalized Fréchet Geometric distribution with parameters η, κ, ϕ, δ, and τ∈0,1 are defined, respectively, as follows:(1)fy;θ¯=1−τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ2,y≥0,η,κ,ϕ,δ>0,τ∈0,1,and(2)Fy;θ¯=1−1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ.θ⊆ℝ5 given by θ¯=η,κ,ϕ,δ,τ∈ℝ5 [19].


## 2. Structural Properties

According to Rafique and Saud [19], the hazard rate, reverse hazard rate, and survival function of the EGFG are defined, respectively, by(3)hy=κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ,(4)ry=1−τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,(5)Sy=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ.The mills ratio and elasticity function of any probability distribution can be defined as(6)my=1−Fyfy,ey=yry.[20] The mills ratio and elasticity function of the EGFG distribution are given by(7)my=1−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1,(8)ey=1−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ.


## 3. Materials and Methods

### 3.1. Characterization via Ratio of Two Truncated Moments

The EGFG distribution is characterized through ratio of two truncated moments. Glanzel [21] stated that this characterization is stable in the sense of weak convergence.Proposition 1.
SupposeY:Ω⟶0,∞ be a continuous random variable with probability density function defined in equation (1) and let(9)k1y=1,(10)k2y=k1y1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0,and then, the function Σ which is presented in theorem (Glänzel [1, 2]) has the following expression:(11)Σy=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0.Proof.
SupposeY has pdf in equation (1), then(12)1−FyEk1|Y≥y=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0,1−FyEk2|Y≥y=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ2,and so(13)Σy=1−FyEk2|Y≥y1−FyEk1|Y≥y=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,(14)Σ′y=−121−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ2.
As(15)Σyk1y−k2y=−k1y21−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ≠0,∀y,the proof is as follows:
Conversely, ifΣ is defined as above, then(16)s′y=Σ′yk1yΣyk1y−k2y=Σ′yk1y−1/2k2y,s′y=κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ,and hence,(17)sy=−ln1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,and(18)e−sy=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0.
Therefore, in the light of theorem, given in Glänzel [1] and Glanzel [21], Y has pdf in equation (1).Corollary 1.
Suppose that the continuous randomY:Ω⟶0,∞ and let k1y as given in proposition equation (1). The pdf of Y in equation (1) provided functions k2y and Σ defined in the theorem (Glänzel [1]). The theorem is based on truncated moments specified in the ratio form.(19)Σ′yk1yΣyk1y−k2y=κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ.Remark 1.
The solution of the equation in corollary equation (1) is(20)Σy=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ∫−gy1−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ2dy+D,where D is the constant.


#### 3.1.1. Special Cases of the Exponentiated Generalized Fréchet Geometric Distribution

The expressions ofk1y, k2y, and Σy to characterize the submodels of the EGFG distribution are given as follows:(1) Generalized Fréchet Geometric Distribution. Puttingκ=1, suppose that(21)k1y=1,k2y=k1y1−τ1−e−δη/yϕ1−τ1−e−δη/yϕ,y≥0,so that Y follows GFG distribution if and only if the function Σ obtained in Proposition 1 has the following expression:(22)Σy=121−τ1−e−δη/yϕ1−τ1−e−δη/yϕ.(2) Fréchet Geometric Distribution. Forκ=δ=1, let(23)k1y=1,k2y=k1y1−τ1−e−η/yϕ1−τ1−e−η/yϕ,y≥0,and if Σ obtained in Proposition 1 has the expression, then Y follows Fréchet Geometric distribution as follows:(24)Σy=121−τ1−e−η/yϕ1−τ1−e−η/yϕ.(3) Inverse Rayleigh Geometric Distribution. Withκ=1, δ=1, andϕ=2, we get(25)k1y=1,k2y=k1y1−τ1−e−η/y21−τ1−e−η/y2,y≥0,and then, r.v. Y follows IRG distribution if the function Σ has the following expression:(26)Σy=121−τ1−e−η/y21−τ1−e−η/y2.(4) Inverse Exponential Distribution. Forκ=δ=ϕ=1, we have(27)k1y=1,k2y=k1y1−τ1−e−η/y11−τ1−e−η/y1,y≥0,and(28)Σy=121−τ1−e−η/y11−τ1−e−η/y1.


### 3.2. Characterization of EGFG under Hazard Rate Function

In this subsection, the characterization of the EGFG distribution is obtained by hazard rate function.Definition 1.
Suppose the absolutely continuous random variable “Y” with pdffy and cdfFy. The hazard function to F is symbolized as hF and is obtained by(29)hFy=fy1−Fy,y∈SuppF,where F represents twice “differentiable distribution function.” The hazard function satisfies the following differential equation:(30)hF′yhFy−hFy=f′yfy=σy,where σy is an appropriate integrable function. The purpose here is to define a differential equation that has a simple form as possible and a nontrivial form as in equation (30). However, in some continuous distributions, this may be impossible.Proposition 2.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of “Y” is equation (1) if and only if its hazard rate function hFy satisfies the following differential equation:(31)hF′y−−ϕ−1y−1hFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has the hrf given in equation (3), then(32)hF′y=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ,hF′y−−ϕ−1y−1hFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ,ddyyϕ+1hFy=ddyκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ,and simplification results in equation (3).


### 3.3. Characterization of EGFG under Reverse Hazard Rate Function

In this subsection, the EGFG distribution is characterized via the reverse hazard rate function.Definition 2.
Suppose the absolutely continuous random variable “Y” with pdf equation (1) and cdf equation (2). The reverse hazard rate function to F is symbolized as rF and is given by(33)rFy=fyFy,y∈SuppF,where F represents the twice “differentiable distribution function.” The reverse hazard function satisfies the following differential equation:(34)f′yfy=rF′yrFy+rFy.Proposition 3.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its reverse hazard function rFy satisfies the following differential equation:(35)rF′y−−ϕ−1y−1rFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ++ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has density function in equation (1), then surely equation (35) holds.(36)rF′y=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδrF′y−−ϕ−1y−1rFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ++ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ,ddyyϕ+1ry=ddy1−τκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,and simplification results in equation (4).


### 3.4. Characterization of EGFG under Mills Ratio

In this subsection, the EGFG distribution is characterized via mills ratio.Definition 3.
Suppose the absolutely continuous random variable “Y” with pdf f(y) and cdf F(y). The mills ratio to F is symbolized as mF and is obtained by(37)mFy=1−Fyfy,y∈SuppF,where F represents the twice “differentiable distribution function.” The mills ratio satisfies the following differential equation:(38)f′yfy=σy=−mF′ymFy+1mFy,where σy is an appropriate integrable function.Proposition 4.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its mills ratio mFy defined in equation (7) satisfies the following differential equation:(39)y−ϕ+1m′y−ϕ+1y−ϕ+2my=−y−ϕ+11−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−ηyϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−y−ϕ+1.Proof.
If the r.vY has density function equation (1), then surely equation (7) holds.(40)m′y=−1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−1,y−ϕ+1m′y−ϕ+1y−ϕ+2my=−y−ϕ+11−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−y−ϕ+1,ddymFyy−ϕ+1=ddy1−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1,and simplification results in equation (7).


### 3.5. Characterization of EGFG under Elasticity Function

In this subsection, the EGFG distribution is characterized through the elasticity function.Definition 4.
Suppose the absolutely continuous random variable “Y” with pdf f(y) and cdf F(y). The elasticity function to F is symbolized as eF and is obtained by(41)eFy=yfyFy,y∈SuppF,where F represents the twice “differentiable distribution function.” The elasticity function satisfies the following equation:(42)f′yfy=eF′yeFy+eFyY−1Y.Proposition 5.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its elasticity function eFy satisfies the following differential equation:(43)eF′y−y−1eFy=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has density function equation (1), then surely equation (43) holds.(44)eF′y=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ+1−τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,eF′y−y−1eFy=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.
Now, the result is as follows: conversely if equation (43) holds, then(45)eFy=1−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,which is the elasticity function of the EGFG distribution.


## 3.1. Characterization via Ratio of Two Truncated Moments

The EGFG distribution is characterized through ratio of two truncated moments. Glanzel [21] stated that this characterization is stable in the sense of weak convergence.Proposition 1.
SupposeY:Ω⟶0,∞ be a continuous random variable with probability density function defined in equation (1) and let(9)k1y=1,(10)k2y=k1y1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0,and then, the function Σ which is presented in theorem (Glänzel [1, 2]) has the following expression:(11)Σy=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0.Proof.
SupposeY has pdf in equation (1), then(12)1−FyEk1|Y≥y=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0,1−FyEk2|Y≥y=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ2,and so(13)Σy=1−FyEk2|Y≥y1−FyEk1|Y≥y=121−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,(14)Σ′y=−121−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ2.
As(15)Σyk1y−k2y=−k1y21−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ≠0,∀y,the proof is as follows:
Conversely, ifΣ is defined as above, then(16)s′y=Σ′yk1yΣyk1y−k2y=Σ′yk1y−1/2k2y,s′y=κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ,and hence,(17)sy=−ln1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,and(18)e−sy=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ,y>0.
Therefore, in the light of theorem, given in Glänzel [1] and Glanzel [21], Y has pdf in equation (1).Corollary 1.
Suppose that the continuous randomY:Ω⟶0,∞ and let k1y as given in proposition equation (1). The pdf of Y in equation (1) provided functions k2y and Σ defined in the theorem (Glänzel [1]). The theorem is based on truncated moments specified in the ratio form.(19)Σ′yk1yΣyk1y−k2y=κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ.Remark 1.
The solution of the equation in corollary equation (1) is(20)Σy=1−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ∫−gy1−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ2dy+D,where D is the constant.


### 3.1.1. Special Cases of the Exponentiated Generalized Fréchet Geometric Distribution

The expressions ofk1y, k2y, and Σy to characterize the submodels of the EGFG distribution are given as follows:(1) Generalized Fréchet Geometric Distribution. Puttingκ=1, suppose that(21)k1y=1,k2y=k1y1−τ1−e−δη/yϕ1−τ1−e−δη/yϕ,y≥0,so that Y follows GFG distribution if and only if the function Σ obtained in Proposition 1 has the following expression:(22)Σy=121−τ1−e−δη/yϕ1−τ1−e−δη/yϕ.(2) Fréchet Geometric Distribution. Forκ=δ=1, let(23)k1y=1,k2y=k1y1−τ1−e−η/yϕ1−τ1−e−η/yϕ,y≥0,and if Σ obtained in Proposition 1 has the expression, then Y follows Fréchet Geometric distribution as follows:(24)Σy=121−τ1−e−η/yϕ1−τ1−e−η/yϕ.(3) Inverse Rayleigh Geometric Distribution. Withκ=1, δ=1, andϕ=2, we get(25)k1y=1,k2y=k1y1−τ1−e−η/y21−τ1−e−η/y2,y≥0,and then, r.v. Y follows IRG distribution if the function Σ has the following expression:(26)Σy=121−τ1−e−η/y21−τ1−e−η/y2.(4) Inverse Exponential Distribution. Forκ=δ=ϕ=1, we have(27)k1y=1,k2y=k1y1−τ1−e−η/y11−τ1−e−η/y1,y≥0,and(28)Σy=121−τ1−e−η/y11−τ1−e−η/y1.


## 3.1.1. Special Cases of the Exponentiated Generalized Fréchet Geometric Distribution

The expressions ofk1y, k2y, and Σy to characterize the submodels of the EGFG distribution are given as follows:(1) Generalized Fréchet Geometric Distribution. Puttingκ=1, suppose that(21)k1y=1,k2y=k1y1−τ1−e−δη/yϕ1−τ1−e−δη/yϕ,y≥0,so that Y follows GFG distribution if and only if the function Σ obtained in Proposition 1 has the following expression:(22)Σy=121−τ1−e−δη/yϕ1−τ1−e−δη/yϕ.(2) Fréchet Geometric Distribution. Forκ=δ=1, let(23)k1y=1,k2y=k1y1−τ1−e−η/yϕ1−τ1−e−η/yϕ,y≥0,and if Σ obtained in Proposition 1 has the expression, then Y follows Fréchet Geometric distribution as follows:(24)Σy=121−τ1−e−η/yϕ1−τ1−e−η/yϕ.(3) Inverse Rayleigh Geometric Distribution. Withκ=1, δ=1, andϕ=2, we get(25)k1y=1,k2y=k1y1−τ1−e−η/y21−τ1−e−η/y2,y≥0,and then, r.v. Y follows IRG distribution if the function Σ has the following expression:(26)Σy=121−τ1−e−η/y21−τ1−e−η/y2.(4) Inverse Exponential Distribution. Forκ=δ=ϕ=1, we have(27)k1y=1,k2y=k1y1−τ1−e−η/y11−τ1−e−η/y1,y≥0,and(28)Σy=121−τ1−e−η/y11−τ1−e−η/y1.


## 3.2. Characterization of EGFG under Hazard Rate Function

In this subsection, the characterization of the EGFG distribution is obtained by hazard rate function.Definition 1.
Suppose the absolutely continuous random variable “Y” with pdffy and cdfFy. The hazard function to F is symbolized as hF and is obtained by(29)hFy=fy1−Fy,y∈SuppF,where F represents twice “differentiable distribution function.” The hazard function satisfies the following differential equation:(30)hF′yhFy−hFy=f′yfy=σy,where σy is an appropriate integrable function. The purpose here is to define a differential equation that has a simple form as possible and a nontrivial form as in equation (30). However, in some continuous distributions, this may be impossible.Proposition 2.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of “Y” is equation (1) if and only if its hazard rate function hFy satisfies the following differential equation:(31)hF′y−−ϕ−1y−1hFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has the hrf given in equation (3), then(32)hF′y=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ,hF′y−−ϕ−1y−1hFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−121−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+1−τ1−1−1−e−η/yϕκδ−1τ1−1−1−e−η/yϕκδ,ddyyϕ+1hFy=ddyκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδ,and simplification results in equation (3).


## 3.3. Characterization of EGFG under Reverse Hazard Rate Function

In this subsection, the EGFG distribution is characterized via the reverse hazard rate function.Definition 2.
Suppose the absolutely continuous random variable “Y” with pdf equation (1) and cdf equation (2). The reverse hazard rate function to F is symbolized as rF and is given by(33)rFy=fyFy,y∈SuppF,where F represents the twice “differentiable distribution function.” The reverse hazard function satisfies the following differential equation:(34)f′yfy=rF′yrFy+rFy.Proposition 3.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its reverse hazard function rFy satisfies the following differential equation:(35)rF′y−−ϕ−1y−1rFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ++ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has density function in equation (1), then surely equation (35) holds.(36)rF′y=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτδ−1δ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδrF′y−−ϕ−1y−1rFy=τ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ++ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1−1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ,ddyyϕ+1ry=ddy1−τκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,and simplification results in equation (4).


## 3.4. Characterization of EGFG under Mills Ratio

In this subsection, the EGFG distribution is characterized via mills ratio.Definition 3.
Suppose the absolutely continuous random variable “Y” with pdf f(y) and cdf F(y). The mills ratio to F is symbolized as mF and is obtained by(37)mFy=1−Fyfy,y∈SuppF,where F represents the twice “differentiable distribution function.” The mills ratio satisfies the following differential equation:(38)f′yfy=σy=−mF′ymFy+1mFy,where σy is an appropriate integrable function.Proposition 4.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its mills ratio mFy defined in equation (7) satisfies the following differential equation:(39)y−ϕ+1m′y−ϕ+1y−ϕ+2my=−y−ϕ+11−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−ηyϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−y−ϕ+1.Proof.
If the r.vY has density function equation (1), then surely equation (7) holds.(40)m′y=−1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−1,y−ϕ+1m′y−ϕ+1y−ϕ+2my=−y−ϕ+11−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδ−1−21−τ1−1−1−e−η/yϕκδ−1+τ−1δδ−11−1−e−η/yϕκ−δ+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−y−ϕ+1,ddymFyy−ϕ+1=ddy1−τ1−1−1−e−η/yϕκδ1−1−1−e−η/yϕκδκδϕηϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1,and simplification results in equation (7).


## 3.5. Characterization of EGFG under Elasticity Function

In this subsection, the EGFG distribution is characterized through the elasticity function.Definition 4.
Suppose the absolutely continuous random variable “Y” with pdf f(y) and cdf F(y). The elasticity function to F is symbolized as eF and is obtained by(41)eFy=yfyFy,y∈SuppF,where F represents the twice “differentiable distribution function.” The elasticity function satisfies the following equation:(42)f′yfy=eF′yeFy+eFyY−1Y.Proposition 5.
Let the random variableY:Ω∈0,∞ follows the EGFG distribution. The density function of Y is equation (1) if and only if its elasticity function eFy satisfies the following differential equation:(43)eF′y−y−1eFy=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.Proof.
If the r.vY has density function equation (1), then surely equation (43) holds.(44)eF′y=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ+1−τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,eF′y−y−1eFy=yτ2κδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−12τ1−τ−τ1−1−1−e−η/yϕκδ1−τ1−1−1−e−η/yϕκδτ−1δδ−11−1−e−η/yϕκ−δ+−ϕ+1y−1+ϕηϕy−ϕ+1−κ−1ϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕ−1τκδϕηϕy−ϕ+1e−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−1−2y1−τ1−1−1−e−η/yϕκδ−1−y1−τ1−1−1−e−η/yϕκδ−2/τ1−1−1−e−η/yϕκδτ1−τ−1−τ1−1−1−e−η/yϕκδ.
Now, the result is as follows: conversely if equation (43) holds, then(45)eFy=1−τκδϕηϕy−ϕe−η/yϕ1−e−η/yϕκ−11−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−11−τ1−1−1−e−η/yϕκδ−1−τ1−1−1−e−η/yϕκδ,which is the elasticity function of the EGFG distribution.


## 4. Entropy of the Exponentiated Generalized Fréchet Geometric Distribution

Entropy measures have a strength of prediction. Different studies of time series may influence the predictive capacity of entropy measures. According to Liang et al. [22], time series analysis affects many entropy methods. Different entropy methods for measuring the complexity of time series were studied by Chen et al. [23]. According to Yin and Shang [24], entropy is a useful tool for examining time series because it does not include any constraints on the probability distribution. Solís-Montufar et al. [25] stated that several entropy measures can be used for complex time series. Its value may be positive, negative, or zero. A higher value shows randomness, while zero shows the certainty of information.


### 4.1. An Integral Result

The following results indicate that some integrals, including the density of the EGFG distribution, may be defined in terms of the incomplete gamma function.Proposition 6.
Letω>0, fy,θ¯, be specified by equation (1) and(46)Iωθ¯=∫0∞fy;θ¯ωdy.
Then,Iωθ¯ exists if and only if minθ¯>max1−1/ω,0, and it is defined as(47)Iωθ¯=1ϕω+k1/ϕη−ωϕ+1−1YmkjiΓ1ϕωϕ+1−1,where Ymkji=∑m=0∞∑k=0∞∑i=0∞∑j=0∞s+b−1b−1τm1−τκδϕηϕω−1i+j+kκω+j−ωkδi+ω−ωjmi.Proof.
Owing to equation (1), we have(48)Iωθ¯=∫0∞1−τκδϕηϕωy−ωϕ+1e−ωη/yϕ1−e−η/yϕωκ−11−1−e−η/yϕκωδ−11−τ1−1−1−e−η/yϕκδ−2ωdy.
For any positive real numberb and for w<1, we have the binomial and generalized binomial expansion:(49)1−wb−1=∑s=0∞b−1s−1sws,(50)1−w−b=∑s=0∞s+b−1b−1ws.
Applying equations (50) and (49), we get(51)Iωθ¯=Ymkji∫0∞y−ωϕ+1e−ω+kη/yϕdy,=1ϕω+k1/ϕη−ωϕ+1−1YmkjiΓ1ϕωϕ+1−1.
In this study, the interest of Proposition6 is that Iωθ¯ is the major ingredient in the definitions of various entropy measures of the EGFG distribution as derived in the next part.


### 4.2. Various Entropy Measures

In this subsection, some important entropy measures are studied using different techniques. The literature contains many entropies that are defined in Table1 for the general distribution with pdffk;θ. Suppose that ω>0 and ω≠0 as basic assumptions in this general case.Table 1
Submodels from the exponentiated generalized fréchet geometric (EGFG) distribution.
Sr.No.γϕκδτModelspdf1η>0ϕ>0κ=1δ>00<τ<1GFG1−τδϕηϕy−ϕ+1e−δη/yϕ1−τ1−e−δη/yϕ22η>0ϕ>0κ=1δ=10<τ<1FG1−τϕηϕy−ϕ+1e−η/yϕ1−τ1−e−η/yϕ23η>0ϕ=2κ=1δ=10<τ<1IRG21−τη2y−2+1e−η/y21−τ1−e−η/y224η>0ϕ=1κ=1δ=10<τ<1IEG1−τδηy−2e−δη/y1−τ1−e−δη/y2From Table2, we see that the integral ∫−∞∞fk;θωdk is central to obtain the considered entropy measures. Now, we study the entropy measures of the EGFG distribution based on Proposition 6.Table 2
Important entropy measures of a distribution withpdffk;θ at ω.
Name of the entropyReferenceNotationExpressionRényiRényi [26]Rωθ1/1−ωlog∫−∞∞fk;θωdkHavrda and CharvátHavrda and Charvát [27]HCωθ1/21−ω−1∫−∞∞fk;θωdk−1ArimotoArimoto [28]Aωθω/1−ω∫−∞∞fk;θωdk1/ω−1TsallisTsallis [29]Tωθ1/ω−11−∫−∞∞fk;θωdk


#### 4.2.1. Rényi Entropy

Based on Table2, equation (1), and Proposition 6, the Rényi entropy of the EGFG distribution can be defined as(52)Rωθ=11−ωlogIωθ,=11−ωlog1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


#### 4.2.2. Havrda and Charvát Entropy

From Table2, equation (1), and Proposition 6, the Havrda and Charvát entropy of the EGFG distribution is specified by(53)HCωθ=121−ω−1Iωθ−1,=121−ω−11ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1−1.


#### 4.2.3. Arimoto Entropy

Again, from Table2, equation (1), and Proposition 6, the Arimoto entropy of the EGFG distribution is expressed by(54)Aωθ=ω1−ωIωθ1/ω−1,=ω1−ω1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−11/ω−1.


#### 4.2.4. Tsallis Entropy

Based on Table2, equation (1), and Proposition 6, the Tsallis entropy of the EGFG distribution can be presented as(55)Tωθ=1ω−11−Iωθ,=1ω−11−1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


### 4.3. Numerical Results

In this section, the idea of following Al-Babtain et al.’s [30] numerical values for the four different entropy measures of the EGFG distribution is calculated. The findings of all considered entropy measures are presented in Tables 3–7.Table 3
Numerical values of the entropy measures of the EGFG distributionη=2, δ=3, τ=0.8, and ϕ=2.
ηκδτϕωRωθHCωθAωθTωθ2330.820.50.3911.3741.4621.1382.90.4061.4411.5491.1932.50.4771.7621.9971.4622330.821.50.1200.4420.2650.2592.90.1320.4800.2880.2812.50.1820.6460.3910.3782330.8220.0770.3260.1700.1632.90.0880.3680.1930.1842.50.1370.5420.2920.271Table 4
Numerical values of the entropy measures of the EGFG distributionη=2.5, κ=3, τ=0.9, and ϕ=2.5.
ηκδτϕωRωθHCωθAωθTωθ2.531.20.92.50.50.2840.9320.9210.7722.20.3041.0121.0140.8383.20.3161.0581.0680.8762.531.20.92.51.50.0240.0950.0560.0552.20.0430.1640.0970.0963.20.0520.2000.1180.1172.531.20.92.52.5−0.046−0.265−0.109−0.1142.2−0.028−0.154−0.065−0.0673.2−0.018−0.100−0.042−0.043Table 5
Numerical values of the entropy measures of the EGFG distributionη=3, κ=3, δ=1.2, and ϕ=2.
ηκδτϕωRωθHCωθAωθTωθ331.20.520.50.6202.5143.1662.0820.60.5952.3752.9351.9670.80.5161.9582.2791.622331.20.521.50.3761.1990.7520.7020.60.3451.1190.6980.6550.80.2480.8470.5190.496331.20.522.50.3101.0160.5800.4830.60.2770.9540.5310.4110.80.1760.7060.3600.304Table 6
Numerical values of the entropy measures of the EGFG distributionη=3.5, δ=3.9, δ=1.5, and τ=0.7.
ηκδτϕωRωθHCωθAωθTωθ3.53.91.50.71.20.50.7243.1454.3022.6052.20.5111.9352.2451.6033.20.3741.2981.3641.0753.53.91.50.71.21.50.4181.3050.8240.7642.20.2890.9680.5980.5673.20.1760.6260.3790.3673.53.91.50.71.22.50.3421.0730.6280.4622.20.2250.8370.4460.3613.20.1160.5100.2460.220Table 7
Entropies of the EGFG for different parameter combinations.
ηκδτϕωRωθHCωθAωθTωθ2.533.20.852.51.60.0930.3540.2050.2012.22.930.82.31.50.1230.4510.2700.2641.32.52.70.521.490.1240.4540.2730.2671.252.22.50.451.81.450.2170.7520.4630.4481.22.11.70.31.50.50.6522.6993.4852.2361.01.91.50.11.30.50.7983.6385.2843.0131.071.91.50.091.30.50.83.6535.3133.0260.81.71.50.091.30.50.8183.7795.5813.1310.71.551.350.0871.250.470.9875.25910.6144.405Tables3–7 show the numerical values of the entropy measures for the different sets of the parameters. It can be noticed that,(i)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing whenκ and ω are increasing.(ii)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing whenδ and ω are increasing.(iii)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing whenτ and ω are increasing.(iv)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing whenϕ and ω are increasing.(v)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing when all parameter values are decreasing.


## 4.1. An Integral Result

The following results indicate that some integrals, including the density of the EGFG distribution, may be defined in terms of the incomplete gamma function.Proposition 6.
Letω>0, fy,θ¯, be specified by equation (1) and(46)Iωθ¯=∫0∞fy;θ¯ωdy.
Then,Iωθ¯ exists if and only if minθ¯>max1−1/ω,0, and it is defined as(47)Iωθ¯=1ϕω+k1/ϕη−ωϕ+1−1YmkjiΓ1ϕωϕ+1−1,where Ymkji=∑m=0∞∑k=0∞∑i=0∞∑j=0∞s+b−1b−1τm1−τκδϕηϕω−1i+j+kκω+j−ωkδi+ω−ωjmi.Proof.
Owing to equation (1), we have(48)Iωθ¯=∫0∞1−τκδϕηϕωy−ωϕ+1e−ωη/yϕ1−e−η/yϕωκ−11−1−e−η/yϕκωδ−11−τ1−1−1−e−η/yϕκδ−2ωdy.
For any positive real numberb and for w<1, we have the binomial and generalized binomial expansion:(49)1−wb−1=∑s=0∞b−1s−1sws,(50)1−w−b=∑s=0∞s+b−1b−1ws.
Applying equations (50) and (49), we get(51)Iωθ¯=Ymkji∫0∞y−ωϕ+1e−ω+kη/yϕdy,=1ϕω+k1/ϕη−ωϕ+1−1YmkjiΓ1ϕωϕ+1−1.
In this study, the interest of Proposition6 is that Iωθ¯ is the major ingredient in the definitions of various entropy measures of the EGFG distribution as derived in the next part.


## 4.2. Various Entropy Measures

In this subsection, some important entropy measures are studied using different techniques. The literature contains many entropies that are defined in Table1 for the general distribution with pdffk;θ. Suppose that ω>0 and ω≠0 as basic assumptions in this general case.Table 1
Submodels from the exponentiated generalized fréchet geometric (EGFG) distribution.
Sr.No.γϕκδτModelspdf1η>0ϕ>0κ=1δ>00<τ<1GFG1−τδϕηϕy−ϕ+1e−δη/yϕ1−τ1−e−δη/yϕ22η>0ϕ>0κ=1δ=10<τ<1FG1−τϕηϕy−ϕ+1e−η/yϕ1−τ1−e−η/yϕ23η>0ϕ=2κ=1δ=10<τ<1IRG21−τη2y−2+1e−η/y21−τ1−e−η/y224η>0ϕ=1κ=1δ=10<τ<1IEG1−τδηy−2e−δη/y1−τ1−e−δη/y2From Table2, we see that the integral ∫−∞∞fk;θωdk is central to obtain the considered entropy measures. Now, we study the entropy measures of the EGFG distribution based on Proposition 6.Table 2
Important entropy measures of a distribution withpdffk;θ at ω.
Name of the entropyReferenceNotationExpressionRényiRényi [26]Rωθ1/1−ωlog∫−∞∞fk;θωdkHavrda and CharvátHavrda and Charvát [27]HCωθ1/21−ω−1∫−∞∞fk;θωdk−1ArimotoArimoto [28]Aωθω/1−ω∫−∞∞fk;θωdk1/ω−1TsallisTsallis [29]Tωθ1/ω−11−∫−∞∞fk;θωdk


### 4.2.1. Rényi Entropy

Based on Table2, equation (1), and Proposition 6, the Rényi entropy of the EGFG distribution can be defined as(52)Rωθ=11−ωlogIωθ,=11−ωlog1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


### 4.2.2. Havrda and Charvát Entropy

From Table2, equation (1), and Proposition 6, the Havrda and Charvát entropy of the EGFG distribution is specified by(53)HCωθ=121−ω−1Iωθ−1,=121−ω−11ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1−1.


### 4.2.3. Arimoto Entropy

Again, from Table2, equation (1), and Proposition 6, the Arimoto entropy of the EGFG distribution is expressed by(54)Aωθ=ω1−ωIωθ1/ω−1,=ω1−ω1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−11/ω−1.


### 4.2.4. Tsallis Entropy

Based on Table2, equation (1), and Proposition 6, the Tsallis entropy of the EGFG distribution can be presented as(55)Tωθ=1ω−11−Iωθ,=1ω−11−1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


## 4.2.1. Rényi Entropy

Based on Table2, equation (1), and Proposition 6, the Rényi entropy of the EGFG distribution can be defined as(52)Rωθ=11−ωlogIωθ,=11−ωlog1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


## 4.2.2. Havrda and Charvát Entropy

From Table2, equation (1), and Proposition 6, the Havrda and Charvát entropy of the EGFG distribution is specified by(53)HCωθ=121−ω−1Iωθ−1,=121−ω−11ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1−1.


## 4.2.3. Arimoto Entropy

Again, from Table2, equation (1), and Proposition 6, the Arimoto entropy of the EGFG distribution is expressed by(54)Aωθ=ω1−ωIωθ1/ω−1,=ω1−ω1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−11/ω−1.


## 4.2.4. Tsallis Entropy

Based on Table2, equation (1), and Proposition 6, the Tsallis entropy of the EGFG distribution can be presented as(55)Tωθ=1ω−11−Iωθ,=1ω−11−1ϕω+k1/ϕη−ωϕ+1+1YmkjiΓ1ϕωϕ+1−1.


## 4.3. Numerical Results

In this section, the idea of following Al-Babtain et al.’s [30] numerical values for the four different entropy measures of the EGFG distribution is calculated. The findings of all considered entropy measures are presented in Tables 3–7.Table 3
Numerical values of the entropy measures of the EGFG distributionη=2, δ=3, τ=0.8, and ϕ=2.
ηκδτϕωRωθHCωθAωθTωθ2330.820.50.3911.3741.4621.1382.90.4061.4411.5491.1932.50.4771.7621.9971.4622330.821.50.1200.4420.2650.2592.90.1320.4800.2880.2812.50.1820.6460.3910.3782330.8220.0770.3260.1700.1632.90.0880.3680.1930.1842.50.1370.5420.2920.271Table 4
Numerical values of the entropy measures of the EGFG distributionη=2.5, κ=3, τ=0.9, and ϕ=2.5.
ηκδτϕωRωθHCωθAωθTωθ2.531.20.92.50.50.2840.9320.9210.7722.20.3041.0121.0140.8383.20.3161.0581.0680.8762.531.20.92.51.50.0240.0950.0560.0552.20.0430.1640.0970.0963.20.0520.2000.1180.1172.531.20.92.52.5−0.046−0.265−0.109−0.1142.2−0.028−0.154−0.065−0.0673.2−0.018−0.100−0.042−0.043Table 5
Numerical values of the entropy measures of the EGFG distributionη=3, κ=3, δ=1.2, and ϕ=2.
ηκδτϕωRωθHCωθAωθTωθ331.20.520.50.6202.5143.1662.0820.60.5952.3752.9351.9670.80.5161.9582.2791.622331.20.521.50.3761.1990.7520.7020.60.3451.1190.6980.6550.80.2480.8470.5190.496331.20.522.50.3101.0160.5800.4830.60.2770.9540.5310.4110.80.1760.7060.3600.304Table 6
Numerical values of the entropy measures of the EGFG distributionη=3.5, δ=3.9, δ=1.5, and τ=0.7.
ηκδτϕωRωθHCωθAωθTωθ3.53.91.50.71.20.50.7243.1454.3022.6052.20.5111.9352.2451.6033.20.3741.2981.3641.0753.53.91.50.71.21.50.4181.3050.8240.7642.20.2890.9680.5980.5673.20.1760.6260.3790.3673.53.91.50.71.22.50.3421.0730.6280.4622.20.2250.8370.4460.3613.20.1160.5100.2460.220Table 7
Entropies of the EGFG for different parameter combinations.
ηκδτϕωRωθHCωθAωθTωθ2.533.20.852.51.60.0930.3540.2050.2012.22.930.82.31.50.1230.4510.2700.2641.32.52.70.521.490.1240.4540.2730.2671.252.22.50.451.81.450.2170.7520.4630.4481.22.11.70.31.50.50.6522.6993.4852.2361.01.91.50.11.30.50.7983.6385.2843.0131.071.91.50.091.30.50.83.6535.3133.0260.81.71.50.091.30.50.8183.7795.5813.1310.71.551.350.0871.250.470.9875.25910.6144.405Tables3–7 show the numerical values of the entropy measures for the different sets of the parameters. It can be noticed that,(i)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing whenκ and ω are increasing.(ii)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing whenδ and ω are increasing.(iii)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing whenτ and ω are increasing.(iv)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing whenϕ and ω are increasing.(v)
Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing when all parameter values are decreasing.


## 5. Conclusion

In this article, the characterization results for the Exponentiated Generalized Fréchet Geometric distribution through five approaches are presented. In the previous section, the analytical expressions of four entropy measures for the EGFG distribution are investigated. All entropy measures are compared numerically by considering the various combination of the parametric values. We hope these expressions of entropies’ measures by various techniques may be helpful in uncertainty measures of a random variable of related fields.



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*Source: 2717894-2022-02-04.xml*