# On the Properties of the New Generalized Pareto Distribution and Its Applications

Authors: Najma Salahuddin; Alamgir Khalil; Wali Khan Mashwani; Sharifah Alrajhi; Sanaa Al-Marzouki; Kamal Shah
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In this paper, a new generalization of the Generalized Pareto distribution is proposed using the generator suggested in [1], named as Khalil Extended Generalized Pareto (KEGP) distribution. Various shapes of the suggested model and important mathematical properties are investigated that includes moments, quantile function, moment-generating function, measures of entropy, and order statistics. Parametric estimation of the model is discussed using the technique of maximum likelihood. A simulation study is performed for the assessment of the maximum likelihood estimates in terms of their bias and mean squared error using simulated sample estimates. The practical applications are illustrated via two real data sets from survival and reliability theory. The suggested model provided better fits than the other considered models.


---

## Body

## 1. Introduction

In modelling heavy-tailed data sets, the Generalized Pareto distribution is a very significant distribution with its applications in environmental studies, finance, operation risk, and insurance. The Generalized Pareto distribution was first familiarized while making inferences on the upper tail of a distribution by Pickands [2]. The distribution is sometimes termed as the “peaks over thresholds” model as it is used for modelling exceedances over threshold level in flood control. In particular, the Generalized Pareto distribution is used to model extreme values. This application of the Generalized Pareto distribution was debated by several authors, for instance, Gupta et al. [3], Hogg et al. [4], Hosking and Wallis [5], Smith [6–8], Davison [9], and Davison and Smith [10]. Smith [8] presented an excellent review of the two most widely used methods in this field, based on generalized extreme value distributions and on the Generalized Pareto distribution. Davison and Smith [10] discussed the applications of Generalized Pareto distribution using river-flow exceedances and used it as a model for excesses over thresholds. Its applications include the upper atmosphere ozone levels, environmental extreme events, large fluctuation in financial data, large insurance claims, and reliability studies. The applications of Generalized Pareto distribution are addressed in various books, such as by Castillo et al. [11] and Kotz and Nadarajah [12]. A number of researchers fit the Generalized Pareto distribution for exceedances over a series of thresholds and have used the Kolmogorov–Smirnov and Anderson–Darling statistics for testing the fit. Various authors claimed the flexibility in modelling long tail data using the Generalized Pareto distribution that includes Choulakian and Stephens [13]. This provides the motivation for proposing another flexible generalization of the Generalized Pareto distribution, referred to as the Khalil Extended Generalized Pareto (KEGP) distribution. The suggested generalization of Generalized Pareto aims the following:(i)
To produce a flexible extension of the Generalized Pareto distribution(ii)
To adapt various forms of the hazard function such as increasing, decreasing, and upside-down failure rate functions(iii)
To produce more flexibility in modelling extreme value data(iv)
To extend the considered distribution to a variety of data related to reliability and survival analysisThe rest of the manuscript is organized as follows: In Section2, the KEGP distribution is defined with a special case of the distribution. In Section 3, some important properties are investigated for instance moments, moment-generating function, entropies, order statistics, and quantile function for the KEGP model. Section 4 is devoted to parametric estimation using the maximum likelihood method. In Section 5, application of the KEGP model is provided using bladder cancer patients’ data and average wind speed data sets. The simulation study is also performed for the parameters using the Monte Carlo simulation method. The paper is concluded finally with some remarks on the results and their significance in Section 6.


## 2. Khalil Extended Generalized Pareto (KEGP)

Salahuddin et al. [1] proposed a very flexible family of distribution, known as the Khalil new generalized family, whose CDF is defined as(1)Gx;α,β,ζ=exp−αFx;ζβ−1exp−α−1,ζ,α,β>0.Pickands [2] suggested the Generalized Pareto distribution with the cumulative distribution function and probability density function in the following form:(2)Fx;δ,ξ=1−1+ξxδ−1/ξ,(3)fx;δ,ξ=1δ1+ξxδ−1/ξ−1.In the above density,δandξ are the scale and shape parameters. Also, forξ>0,thenx≥0; forξ<0,then0≤x≤−δ/ξ; and for ξ=0, the density reduces to exponential distribution.Now, we shall define the Khalil Extended Generalized Pareto distribution using equation (2) in the generator defined in equation (1), that is, if the Generalized Pareto distribution is distributed as x∼GPDξ,δ, then the Khalil Extended Generalized Pareto is distributed as x∼KEGPα,β,ξ,δ with a cumulative distribution function defined as(4)Gx;α,β,ξ,δ=e−α1−1+ξx/δ−1/ξβ−1e−α−1,x≥0,α>0,β>0,δ>0,ξ>0.The probability density function, reliability function, and hazard function are(5)gx;α,β,ξ,δ=αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ,x≥0,α>0,β>0,ξ,δ>0,(6)Rx;α,β,ξ,δ=1−e−α1−1+ξx/δ−1/ξβ−1e−α−1,(7)hx;α,β,ξ,δ=αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αe−α−1δe−α1−1+ξx/δ−1/ξβ−1.Figures1–3 demonstrate the graphs of CDF, PDF, and hazard rate functions, respectively. From Figure 2, it can be clearly seen that as the values of the shape parameters α and β increase (i.e., greater than 2), the pdf of the KEGP approaches more towards normality. Furthermore, in general, the Generalized Pareto distribution has a decreasing hazard function for all values of x greater than 0, while the hazard function illustrated in Figure 3 is very flexible depicting various shapes (i.e., monotonically increasing and decreasing).Figure 1
CDF of the KEGP distribution with various values of the parameters.Figure 2
PDF of the KEGP distribution with various values of the parameters.Figure 3
Hazard function of the KEGP distribution with various values of the parameters.


### 2.1. Special Case

As a special case if we substituteξ=0, in equations (4) and (5), it shall refer to the probability density function and cumulative distribution function of the Exponentiated Exponential Poisson (EEP) [14] as(8)Gx=exp−α1−exp−λxβ−11−exp−α,λ>0,α,β>0,x>0,gx=αβλexp−α1−exp−λxβ−λx1−exp−λxβ−11−exp−α.


## 2.1. Special Case

As a special case if we substituteξ=0, in equations (4) and (5), it shall refer to the probability density function and cumulative distribution function of the Exponentiated Exponential Poisson (EEP) [14] as(8)Gx=exp−α1−exp−λxβ−11−exp−α,λ>0,α,β>0,x>0,gx=αβλexp−α1−exp−λxβ−λx1−exp−λxβ−11−exp−α.


## 3. Statistical Properties of KEGPD

The statistical properties of the KEGP distribution are as follows.


### 3.1. Moments

Let a random variablex follow the KEGP distribution; then, the rth moment of the KEGP distribution, say μr, by definition is(9)μr=Exr=∫0∞xrgx;α,β,ξ,δdx,μr=Exr=∫0∞xrαβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδdx,μr=αβ1−e−αδ∫0∞xr1+ξxδ−1/ξ−11−1+ξxδ−1/ξβ−1e−α1−1+ξx/δ−1/ξβdx.Using the exponent series asex=∑m=0∞xm/m! and the Binomial expansion 1−zb=∑l=0∞−1blzl, on simplification, equation (9) yields(10)μr=αβ1−e−αδ∑i=0∞−αii!∫0∞xr1+ξxδ−1/ξ−1∑l=0∞−1lβ+βi−1l1+ξxδ−1/ξdx,μr=αβ1−e−αδ∑i=0∞−αii!∑l=0∞−1lβ+βi−1l∫0∞xr1+ξxδ−l/ξ−1/ξ−1dx.Now, here we will proof the aboverth moment for three cases as follows:Case-I: whenξ<0, then 0<x<−δ/ξ and the expression in equation (10) can be simplified asIf we suppose−δ/ξx=t,thenx=−ξ/δt,dx=−ξ/δdt,and0<t<1,(11)μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l∫01−ξδtr1−t−l/ξ−1/ξ−1−ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1∫01−ξδtr1−t−l/ξ−1/ξ−1−ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1Br+1,−lξ−1ξ,−1<r<∞,whereBx,y=∫01tx−11−ty−1dt.Case-II: whenξ>0, then 0<x<∞ and the expression in equation (11) can be simplified asNow, here if we supposeδ/ξx=t,thenx=ξ/δt,dx=ξ/δdt,and0<t<∞,(12)μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l∫0∞ξδtr1+t−l/ξ−1/ξ−1ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1lξδr+1∫0∞tr+1−11+t−l/ξ−1/ξ−1dt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1Br+1,lξ+1ξ−r,−1<r<−lξ−1ξ,whereBx,y=∫0∞tx−1/1+tx+ydt of the second kind is used. Now, if we look into the two cases, that is, ξ<0andξ>0, only the second argument of the beta function is different and the first one is the same for both cases.Case-III: ifξ=0, the derivation is straightforward with the density of exponential as(13)μr=αβ1−e−αδ∑i=0∞−αii!∑l=0∞−1lβ+βi−1l∫0∞xre−1/δl+1xdx,μr=αβδr1−e−α∑i=0∞−αii!∑l=0∞−1lβ+βi−1lΓr+1l+1r+1.


### 3.2. Moment-Generating Function (MGF)

If a random variablex has the KEGP x;α,β,ξ,δ, then the MGF of x is obtained by using(14)Mxt=Eetx=∫0∞etxgx;α,β,ξ,δdx.The Taylor series yields the following simplified expression:(15)Mxt=∑r=0∞trr!Exr.Using equations (11), (12), and (14) in (15), we get(16)Mxt=αβ1−e−αδ∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1l−ξδr+1Br+1,−lξ−1ξ,rϵ−1,∞,ξ>0,αβ1−e−αδ∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1l−ξδr+1Br+1,lξ+1ξ−r,rϵ−1,−lξ−1ξ,ξ<0,αβδr1−e−α∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1lΓr+1l+1r+1,rϵ−1,∞,ξ=0.


### 3.3. Entropy Measures

For measuring the randomness of various systems, entropy measures are widely used. The main usage of these entropies is in the areas of physics, sparse kernel density estimation, and molecular imaging of tumors. If the result of entropy statistics is low, it specifies less uncertainty in the data. Thus, the Renyi [15] and q-entropy [16] are considered for the KEGP distribution to measure the quantity of uncertainty in the data. The entropies are characterized as(17)RExρ=1ρ−1log∫−∞∞fρx,Hxq=1q−11−∫−∞∞fqx.Forρ>1, the Renyi entropy of the KEGP distribution is as follows:(18)RExρ=1ρ−1log∫0∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδρdx,=1ρ−1logαβρδρ1−e−αρ∫0∞1+ξxδ−1/ξ+11−1+ξxδ−1/ξβ−1e−α1−1+ξx/δ−1/ξβρdx.On solving and using the exponent series asex=∑m=0∞xm/m! and Binomial expansion as 1−zb=∑l=0∞−1lblzl in expression (18), it becomes(19)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∫0∞1+ξxδ−ρ1/ξ+1∑l=0∞−1lρβ−1+βil1+ξxδ−l/ξdx,=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑l=0∞−1lρβ−1+βil∫0∞1+ξxδ−ρ1/ξ+1.Whenξ<0, then 0<x<−δ/ξ, and on simplification, the Renyi entropy yields the following result:(20)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βil∫0−δ/ξ1+ξxδ−l/ξ−ρ1/ξ+1dx,REx=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βilδξ−ρ1/ξ+1−1/ξ+1.When ξ>0, then 0<x<∞, and on simplification, the Renyi entropy becomes(21)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βil∫0∞1+ξxδ−l/ξ−ρ1/ξ+1dx,REx=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βilδξρ1/ξ+1+l/ξ−1.The q-entropy familiarized by [16] is defined as(22)Hxq=1q−11−∫−∞∞fqx,Hxq=1q−11−∫1∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδq.Consider the integral(23)∫0∞fqx=∫0∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδqdx,for ξ<0; then, 0<x<−δ/ξ, and on simplification, the q-entropy yields the following result:(24)∫0∞fqx=αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξ−q1/ξ+1−l/ξ+1.Using the result of the above integral inHxq, it reduces for ξ<0 as(25)Hxq=1q−11−αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξ−q1/ξ+1−l/ξ+1.and for ξ>0 and 0<x<∞ and on simplification, the integral becomes(26)∫0∞fqx=αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξq1/ξ+1−l/ξ−1.Now, substituting the result inHxq, the q-entropy for ξ>0 is(27)Hxq=1q−11−αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξq1/ξ+1+l/ξ−1.


### 3.4. Quantile Function

The quantile function or random number generator of the KEGP can be obtained by invertingGx;α,β,ξ,δ=u and solving the expression for x as X=Qu=G−1u; we have(28)Q=δ1−−log1−e−a1/1−e−α−u/αβ−ξ−1ξ,where uϵ0,1. When u is replaced by q, the median, 1st quantile, and 3rd quartiles are obtained from (28) by substituting q = 0.5, 0.25, and 0.75, respectively.


### 3.5. Order Statistics

Order statistics plays a vital role in the field of reliability and survival analysis. Let a random samplex¯ of size k from the KEGP distribution has the corresponding order statistics denoted by X1:k<X2:k<X3:k<⋯⋯<Xk:k. Then, the pdf of the rth-order statistics is given by(29)gr:kx=k!r−1!k−r!Gr−1x1−Gxk−rgx,gr:kx=k!r−1!k−r!∑m=0k−r−1mk−rmGm+r−1xgx.Using equations (4) and (5) in (29), we have(30)gr:kx=k!r−1!k−r!∑m=0k−r−1mk−rme−α1−1+ξx/δ−1/ξβ−1e−α−1m+r−1αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ,gr:kx=k!αβr−1!k−r!∑m=0k−r−1mk−rme−α1−1+ξx/δ−1/ξβ−1e−α−1r+m−1αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ.


## 3.1. Moments

Let a random variablex follow the KEGP distribution; then, the rth moment of the KEGP distribution, say μr, by definition is(9)μr=Exr=∫0∞xrgx;α,β,ξ,δdx,μr=Exr=∫0∞xrαβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδdx,μr=αβ1−e−αδ∫0∞xr1+ξxδ−1/ξ−11−1+ξxδ−1/ξβ−1e−α1−1+ξx/δ−1/ξβdx.Using the exponent series asex=∑m=0∞xm/m! and the Binomial expansion 1−zb=∑l=0∞−1blzl, on simplification, equation (9) yields(10)μr=αβ1−e−αδ∑i=0∞−αii!∫0∞xr1+ξxδ−1/ξ−1∑l=0∞−1lβ+βi−1l1+ξxδ−1/ξdx,μr=αβ1−e−αδ∑i=0∞−αii!∑l=0∞−1lβ+βi−1l∫0∞xr1+ξxδ−l/ξ−1/ξ−1dx.Now, here we will proof the aboverth moment for three cases as follows:Case-I: whenξ<0, then 0<x<−δ/ξ and the expression in equation (10) can be simplified asIf we suppose−δ/ξx=t,thenx=−ξ/δt,dx=−ξ/δdt,and0<t<1,(11)μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l∫01−ξδtr1−t−l/ξ−1/ξ−1−ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1∫01−ξδtr1−t−l/ξ−1/ξ−1−ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1Br+1,−lξ−1ξ,−1<r<∞,whereBx,y=∫01tx−11−ty−1dt.Case-II: whenξ>0, then 0<x<∞ and the expression in equation (11) can be simplified asNow, here if we supposeδ/ξx=t,thenx=ξ/δt,dx=ξ/δdt,and0<t<∞,(12)μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l∫0∞ξδtr1+t−l/ξ−1/ξ−1ξδdt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1lξδr+1∫0∞tr+1−11+t−l/ξ−1/ξ−1dt,μr=αβ1−e−αδ∑i=0∞−αii!∑i=0∞−1lβ+βi−1l−ξδr+1Br+1,lξ+1ξ−r,−1<r<−lξ−1ξ,whereBx,y=∫0∞tx−1/1+tx+ydt of the second kind is used. Now, if we look into the two cases, that is, ξ<0andξ>0, only the second argument of the beta function is different and the first one is the same for both cases.Case-III: ifξ=0, the derivation is straightforward with the density of exponential as(13)μr=αβ1−e−αδ∑i=0∞−αii!∑l=0∞−1lβ+βi−1l∫0∞xre−1/δl+1xdx,μr=αβδr1−e−α∑i=0∞−αii!∑l=0∞−1lβ+βi−1lΓr+1l+1r+1.


## 3.2. Moment-Generating Function (MGF)

If a random variablex has the KEGP x;α,β,ξ,δ, then the MGF of x is obtained by using(14)Mxt=Eetx=∫0∞etxgx;α,β,ξ,δdx.The Taylor series yields the following simplified expression:(15)Mxt=∑r=0∞trr!Exr.Using equations (11), (12), and (14) in (15), we get(16)Mxt=αβ1−e−αδ∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1l−ξδr+1Br+1,−lξ−1ξ,rϵ−1,∞,ξ>0,αβ1−e−αδ∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1l−ξδr+1Br+1,lξ+1ξ−r,rϵ−1,−lξ−1ξ,ξ<0,αβδr1−e−α∑r=0∞trr!∑i=0∞−αii!∑l=0∞−1lβ+βi−1lΓr+1l+1r+1,rϵ−1,∞,ξ=0.


## 3.3. Entropy Measures

For measuring the randomness of various systems, entropy measures are widely used. The main usage of these entropies is in the areas of physics, sparse kernel density estimation, and molecular imaging of tumors. If the result of entropy statistics is low, it specifies less uncertainty in the data. Thus, the Renyi [15] and q-entropy [16] are considered for the KEGP distribution to measure the quantity of uncertainty in the data. The entropies are characterized as(17)RExρ=1ρ−1log∫−∞∞fρx,Hxq=1q−11−∫−∞∞fqx.Forρ>1, the Renyi entropy of the KEGP distribution is as follows:(18)RExρ=1ρ−1log∫0∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδρdx,=1ρ−1logαβρδρ1−e−αρ∫0∞1+ξxδ−1/ξ+11−1+ξxδ−1/ξβ−1e−α1−1+ξx/δ−1/ξβρdx.On solving and using the exponent series asex=∑m=0∞xm/m! and Binomial expansion as 1−zb=∑l=0∞−1lblzl in expression (18), it becomes(19)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∫0∞1+ξxδ−ρ1/ξ+1∑l=0∞−1lρβ−1+βil1+ξxδ−l/ξdx,=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑l=0∞−1lρβ−1+βil∫0∞1+ξxδ−ρ1/ξ+1.Whenξ<0, then 0<x<−δ/ξ, and on simplification, the Renyi entropy yields the following result:(20)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βil∫0−δ/ξ1+ξxδ−l/ξ−ρ1/ξ+1dx,REx=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βilδξ−ρ1/ξ+1−1/ξ+1.When ξ>0, then 0<x<∞, and on simplification, the Renyi entropy becomes(21)=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βil∫0∞1+ξxδ−l/ξ−ρ1/ξ+1dx,REx=1ρ−1logαβρδρ1−e−αρ∑i=0∞−αρii!∑i=0∞−1lρβ−1+βilδξρ1/ξ+1+l/ξ−1.The q-entropy familiarized by [16] is defined as(22)Hxq=1q−11−∫−∞∞fqx,Hxq=1q−11−∫1∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδq.Consider the integral(23)∫0∞fqx=∫0∞αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδqdx,for ξ<0; then, 0<x<−δ/ξ, and on simplification, the q-entropy yields the following result:(24)∫0∞fqx=αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξ−q1/ξ+1−l/ξ+1.Using the result of the above integral inHxq, it reduces for ξ<0 as(25)Hxq=1q−11−αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξ−q1/ξ+1−l/ξ+1.and for ξ>0 and 0<x<∞ and on simplification, the integral becomes(26)∫0∞fqx=αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξq1/ξ+1−l/ξ−1.Now, substituting the result inHxq, the q-entropy for ξ>0 is(27)Hxq=1q−11−αβqδq1−e−αq∑i=0∞−αqii!∑l=0∞−1lqβ−1+βilδξq1/ξ+1+l/ξ−1.


## 3.4. Quantile Function

The quantile function or random number generator of the KEGP can be obtained by invertingGx;α,β,ξ,δ=u and solving the expression for x as X=Qu=G−1u; we have(28)Q=δ1−−log1−e−a1/1−e−α−u/αβ−ξ−1ξ,where uϵ0,1. When u is replaced by q, the median, 1st quantile, and 3rd quartiles are obtained from (28) by substituting q = 0.5, 0.25, and 0.75, respectively.


## 3.5. Order Statistics

Order statistics plays a vital role in the field of reliability and survival analysis. Let a random samplex¯ of size k from the KEGP distribution has the corresponding order statistics denoted by X1:k<X2:k<X3:k<⋯⋯<Xk:k. Then, the pdf of the rth-order statistics is given by(29)gr:kx=k!r−1!k−r!Gr−1x1−Gxk−rgx,gr:kx=k!r−1!k−r!∑m=0k−r−1mk−rmGm+r−1xgx.Using equations (4) and (5) in (29), we have(30)gr:kx=k!r−1!k−r!∑m=0k−r−1mk−rme−α1−1+ξx/δ−1/ξβ−1e−α−1m+r−1αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ,gr:kx=k!αβr−1!k−r!∑m=0k−r−1mk−rme−α1−1+ξx/δ−1/ξβ−1e−α−1r+m−1αβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ.


## 4. Parameter Estimation of KEGP Distribution

The parametric estimation of the KEGP distribution is presented in this section using the technique of maximum likelihood since it possesses many desirable properties, for instance, consistency, invariance, and normal approximation. It basically depends upon the maximization of the likelihood function.


### 4.1. Maximum Likelihood Estimation

Let us consider a random sampleX=x1,x2,…xn from the KEGP (x;α,β,ξ,δ model; then, its likelihood function is defined as(31)lx;α,β,λ=∏i=1nαβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ.The log likelihood function of the KEGP model is obtained by taking the logarithm on both sides of (20) as(32)loglx;α,β,ξ,δ=∑i=1nlogα+∑i=1nlogβ−1ξ+1∑i=1nlog1+ξxδ−∑i=1nα1−1+ξxδ−1/ξβ+β−1⋅∑i=1nlog1−1+ξxδ−1/ξ−∑i=1nlog1−e−α−∑i=0∞δ.For MLEs of the unknown parameters of the KEGP model, the nonlinear equation derived above is simplified by taking its derivative w.r.t. toα,β,ξ,andδ, respectively, and setting loglx;α,β,ξ,δ/∂α=0, loglx;α,β,ξ,δ/∂β=0, loglx;α,β,ξ,δ/∂ξ=0, and loglx;α,β,ξ,δ/∂δ=0. The normal equations are as follows:(33)∂loglx;α,β,ξ,δ∂α=nα−∑i=0n1−1+ξxδ−1/ξβ−∑i=1ne−α1−e−α,∂loglx;α,β,ξ,δ∂β=nβ−α∑i=1n1−1+ξxδ−1/ξβlog1−1+ξxδ−1/ξ+∑i=1nlog1−1+ξxδ−1/ξ,∂loglx;α,β,ξ,δ∂ξ=∑i=1nx1/ξ+1δxξ/δ+1−logxξ/δ+1ξ2+αβ∑i=1n1−1+ξxδ−1ξβ−1logxξ/δ+1ξ2−xδξxξ/δ+11+ξxδ−1/ξ,∂loglx;α,β,ξ,δ∂δ=−nδ+1ξ+1∑i=1nxξδδ+xξ+αβ∑i=1nx1−1+ξx/δ−1/ξβ−11+ξx/δ−1/ξ−1δ2−β−1∑i=0nx1+ξx/δ1/ξ−1δδ+xξ.


### 4.2. Asymptotic Confidence Intervals

Since the expression of the MLEs cannot be derived in closed form, for the solution, some iterative procedures such as conjugate gradient type algorithms may be used to obtain the numerical solution. Thus, the asymptotic confidence interval can be derived for the unknown parametersα,β,ξ,andδ with the assumption that the MLEs (α,β,ξ,δ) of these parameters are approximately normal with mean α,β,ξ,δ and inverse Fisher information-observed variance-covariance matrix FI−1. The 4×4 Fisher information matrix is defined as(34)FI−1=∂2logl∂α2∂2logl∂α∂β∂2logl∂α∂ξ∂2logl∂α∂δ∂2logl∂α∂β∂logl∂β2∂2logl∂β∂ξ∂2logl∂β∂δ∂2logl∂α∂ξ∂2logl∂β∂ξ∂logl∂ξ2∂2logl∂ξ∂δ∂2logl∂α∂δ∂2logl∂β∂δ∂2logl∂ξ∂δ∂logl∂δ2−1,whose elements are listed in “Appendix.”Thus, the asymptotic confidence intervals1−ζ100% for the parameters of the KEGP model are constructed by using the following equations:(35)α^±Zζ/2varα^,ξ^±Zζ/2varξ^,β^±Zζ/2varβ^,δ^±Zζ/2varδ^,where Zζ/2 is the upper ζ/2th percentile of the N0,1.


## 4.1. Maximum Likelihood Estimation

Let us consider a random sampleX=x1,x2,…xn from the KEGP (x;α,β,ξ,δ model; then, its likelihood function is defined as(31)lx;α,β,λ=∏i=1nαβ1+ξx/δ−1/ξ−11−1+ξx/δ−1/ξβ−1e−α1−1+ξx/δ−1/ξβ1−e−αδ.The log likelihood function of the KEGP model is obtained by taking the logarithm on both sides of (20) as(32)loglx;α,β,ξ,δ=∑i=1nlogα+∑i=1nlogβ−1ξ+1∑i=1nlog1+ξxδ−∑i=1nα1−1+ξxδ−1/ξβ+β−1⋅∑i=1nlog1−1+ξxδ−1/ξ−∑i=1nlog1−e−α−∑i=0∞δ.For MLEs of the unknown parameters of the KEGP model, the nonlinear equation derived above is simplified by taking its derivative w.r.t. toα,β,ξ,andδ, respectively, and setting loglx;α,β,ξ,δ/∂α=0, loglx;α,β,ξ,δ/∂β=0, loglx;α,β,ξ,δ/∂ξ=0, and loglx;α,β,ξ,δ/∂δ=0. The normal equations are as follows:(33)∂loglx;α,β,ξ,δ∂α=nα−∑i=0n1−1+ξxδ−1/ξβ−∑i=1ne−α1−e−α,∂loglx;α,β,ξ,δ∂β=nβ−α∑i=1n1−1+ξxδ−1/ξβlog1−1+ξxδ−1/ξ+∑i=1nlog1−1+ξxδ−1/ξ,∂loglx;α,β,ξ,δ∂ξ=∑i=1nx1/ξ+1δxξ/δ+1−logxξ/δ+1ξ2+αβ∑i=1n1−1+ξxδ−1ξβ−1logxξ/δ+1ξ2−xδξxξ/δ+11+ξxδ−1/ξ,∂loglx;α,β,ξ,δ∂δ=−nδ+1ξ+1∑i=1nxξδδ+xξ+αβ∑i=1nx1−1+ξx/δ−1/ξβ−11+ξx/δ−1/ξ−1δ2−β−1∑i=0nx1+ξx/δ1/ξ−1δδ+xξ.


## 4.2. Asymptotic Confidence Intervals

Since the expression of the MLEs cannot be derived in closed form, for the solution, some iterative procedures such as conjugate gradient type algorithms may be used to obtain the numerical solution. Thus, the asymptotic confidence interval can be derived for the unknown parametersα,β,ξ,andδ with the assumption that the MLEs (α,β,ξ,δ) of these parameters are approximately normal with mean α,β,ξ,δ and inverse Fisher information-observed variance-covariance matrix FI−1. The 4×4 Fisher information matrix is defined as(34)FI−1=∂2logl∂α2∂2logl∂α∂β∂2logl∂α∂ξ∂2logl∂α∂δ∂2logl∂α∂β∂logl∂β2∂2logl∂β∂ξ∂2logl∂β∂δ∂2logl∂α∂ξ∂2logl∂β∂ξ∂logl∂ξ2∂2logl∂ξ∂δ∂2logl∂α∂δ∂2logl∂β∂δ∂2logl∂ξ∂δ∂logl∂δ2−1,whose elements are listed in “Appendix.”Thus, the asymptotic confidence intervals1−ζ100% for the parameters of the KEGP model are constructed by using the following equations:(35)α^±Zζ/2varα^,ξ^±Zζ/2varξ^,β^±Zζ/2varβ^,δ^±Zζ/2varδ^,where Zζ/2 is the upper ζ/2th percentile of the N0,1.


## 5. Applications

The applications of the KEGP distribution are provided in the following section using simulated data and two real data sets.


### 5.1. Simulation Study

A simulation study is performed to obtain the average values of MLEs (maximum likelihood estimators), MSEs (mean square error), and bias. The following steps are used to perform the simulation.Step 1.
First, suppose the values of the parameters from KEGP asα,β,ξ,δ, that is, Case I = (2, 3, 2, 1), Case II = (3, 4, 2, 1), Case III = (1, 2, 4, 3), and Case IV = (1, 2, 2, 5).Step 2.
The process is repeated 10,000 times and the MSE and bias are computed for the estimates forn = 50, 100, 500, and 1000.Step 3.
Use the following quantile expression for the generation of random numbers from KEGP as(36)X=δ1−−log1−e−a1/1−e−α−u/αβ−ξ−1ξ,where U∼0,1.
Also, the bias and MSEs are calculated using the following expression:(37)MSE=1w∑i=0wbi^−b,bias=1w∑i=0wbi^−b2,where b=α,β,ξ,δ. Results of simulations were obtained for different combinations of the parameters α,β,ξ,andδ and are displayed in Table 1. It is exhibited clearly from the table that with the increase of sample size, these estimates are reasonably consistent and approaches the true values of parameters. Furthermore, the MSEs and bias decrease for all the combinations of parameters with increase of the sample size. Therefore, it has been determined that the process of MLE is performing well in estimating the parameters of KEGPD.Table 1
Average values of MLEs, MSEs, and bias.
Parametersnα^β^ξ^δ^MSE (α^)MSE (β^)MSE (ξ^)MSE (δ)Bias (α^)Bias (β^)Bias (ξ^)Bias (δ^)α = 2β = 3ξ=2δ=1502.82244.3792.38620.88919.74134.83441.73911.46040.82241.37900.3861−0.11081003.12283.6622.53941.317312.4443.059022.79103.18061.12270.66210.53940.317315002.29473.5612.16990.92981.95011.66310.41030.28840.29470.56080.1699−0.0701910002.17203.5672.09770.87340.83161.48080.15830.187220.17200.56710.0977−0.1265α = 3β = 4ξ=2δ=1504.04084.04852.51811.721923.36803.52192.90164.7391.04080.04850.51810.72191003.56144.59612.21831.422511.01024.47911.47472.30680.56140.59610.21830.42255002.76754.76891.93650.92591.74993.39650.21290.3039−0.23250.7689−0.0634−0.074110002.98274.62981.99920.94570.95942.6280.13120.2547−0.01730.6298−0.0007−0.0543α = 1β = 2ξ=4δ=3502.03672.5255.23044.239023.97112.885612.22017.4771.03670.52541.23041.23901002.29802.5495.35303.913011.14341.505512.891220.74281.29800.549241.35300.91305001.34292.3724.28493.41991.17441.06081.09587.52080.34300.37240.28490.4198910001.15632.3644.10103.00190.34200.91780.16504.08660.15630.36400.10100.0019α = 1β = 2ξ=2δ=5503.1713.26783.32614.514434.13795.760410.722419.58792.17111.26781.3261−0.48551002.5453.07133.2014.87629.0924.6516.118725.2251.54521.07131.2011−0.12435002.21872.5112.79854.80929.60561.49143.242010.34091.21870.51100.7985−0.190810001.68322.28122.4574.7914.41610.46601.55545.37460.68320.28120.4569−0.2091


### 5.2. Real-Life Applications

The practicality of the KEGP is demonstrated by using two real-life data sets. The first data set is considered from [17], and the second data set is taken from [18]. The KEGP is compared with various submodels such as Kumaraswamy Pareto [19], Alpha Power Pareto [20], Generalized Pareto [2], and Exponentiated Generalized Pareto [21] distributions with the following probability density functions:(i)
Kumaraswamy Pareto (KP) distribution:(38)fx=abkβkxk+11−βxka−11−1−βxkab−1,x≥β,a,b,k>0.(ii)
Alpha Power Pareto (APP):(39)fx=βlogαα−1α1−x−βx−β−1,α≠0,x>0,fx,α=0,x>0.(iii)
Generalized Pareto (GP):(40)fx=1δ1+ξxδ−1/ξ−1,ξ≠0,x≥0,δ>0.(iv)
Exponentiated Generalized Pareto (EGP):(41)fx=exδ1+ξexδ−1/ξ−1,ξ≠0,−∞<x<∞,δ>0.The performance of the KEGP is checked by comparing it with various forms of Pareto distributions discussed above using the goodness-of-fit test criteria. A package of R software, that is, Adequacy Model, is utilized which includes the outcome of Akaike’s Information Criteria (AIC), Bayesian Information Criterion (BIC), Consistent Akaike’s Information Criteria (CAIC), Hannan–Quinn Information Criteria (HQIC),−lnθ^, and Kolmogorov–Smirnov test (KS) and its p value. Generally, the model is considered as the best-fitted model to the data if the values of all the above criteria are smaller with a larger p value. Tables 2 and 3 display the MLEs and Kolmogorov–Smirnov (K-S) results along with their p values. It can be seen clearly that the results of the Khalil Extended Generalized Pareto distribution for the K-S test are smaller with a large p value, i.e., K-S value = 0.0342, p value = 0.9982 for data set I and K-S value = 0.0835, p value = 0.4881 for data set II, in comparison with its other variants, hence proving a better fit. Furthermore, Tables 4 and 5 display the results of AIC, BIC, CAIC, and HQIC for which the Khalil Extended Generalized Pareto distribution has lower values in comparison with its other various versions. A commendable performance of the KEGP distribution is also visible from Figures 4 and 5 displaying the empirical and theoretical pdfs and CDFs. Figures 6 and 7 demonstrate the Q-Q and P-P plots for both the data sets, respectively. The better performance of the suggested model indicates the need of introduction of new distributions in managing some sets of real data.Table 2
MLEs and standard errors (in parenthesis), K-S values, andp values of data set I.
ModelMLEs of the parameterK-Sp valuesKEGPDα,β,ξ,δ−3.8740, 0.7348, 0.30490, 3.5039 (2.111, 0.519, 0.0891, 0.950)0.03420.9982KP (β,a,b,k)0.05109, 0.3212, 7.1078, 4.3484 (0.009, 0.042, 1.047, 1.373)0.17540.00075APPα,β1.3800, 0.6023 (0.500, 0.0579)1.8082.2e − 16GPξ,δ8.8108, 0.0647 (1.039, 0.078)0.09410.2064EGPξ,δ20.4552, 8.3284 (6.464, 0.794)0.050020.9059Table 3
MLEs and standard errors (in parenthesis), K-S values, andp values of data set II.
ModelMLEs of the parameterK-Sp valuesKEGPDα,β,ξ,δ0.8836, 7.9088, −0.1279, 3.4898 (5.816, 20.854, 0.836, 9.016)0.08350.4881KP (β,a,b,k)1.5574, 0.8636, 7.7916, 9.7403 (0.936, 6.909, 0.179, 3.9484)0.09400.3324APPα,β36.1665, 1.0601 (10.175, 0.068)0.34121.537e − 10GPξ,δ11.6718, −0.7904 (0.583, 0.037)0.24779.291e − 06EGPξ,δ49.9512, 4.0312 (11.410, 0.520)0.26851.086e − 06Table 4
AIC, BIC, CAIC, HQIC, and -L for data set I.
ModelsAICCAICBICHQIC−LKEGPDα,β,ξ,δ830.0632830.3884841.4713834.6984411.0316KP (β,a,b,k)894.9182895.2434906.3263899.5533443.459APPα,β848.9063849.0023854.6103851.2238422.4531GPξ,δ833.7435833.8395839.4476836.0611414.8718EGPξ,δ842.0753842.1713847.7794844.3929419.0377Table 5
AIC, BIC, CAIC, HQIC, and −L for data set II.
ModelsAICCAICBICHQIC−LKEGPDα,β,ξ,δ478.3736478.7947488.7943482.5911235.1868KPβ,a,b,k480.2818480.7029490.7025484.4992236.6409APPα,β614.2206614.3443619.4309616.3293305.1103GPξ,δ537.4365537.5602542.6468539.5452266.7182EGPξ,δ527.2457527.3694532.456529.3544261.6228Figure 4
Density comparison of data set I.Figure 5
Density comparison of data set II.Figure 6
Quantiles and probability plots of data set I.Figure 7
Quantiles and probability plots for data set II.


#### 5.2.1. Data Set I

The first data set is considered from [17], which consists of remission time (in months) of bladder cancer patients. The data set is positively skewed and unimodal, with a skewness of 3.286, mean remission time of 9.366 months, and standard deviation of 10.508 months. These data are recently studied in [22, 23] as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.2, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.4, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.5, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.7, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.26, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.09, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.4, 3.02, 4.43, 5.71, 7.93, 11.79, 18.10, 1.46, 4.4, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.5, 6.25, 8.37, 12.02, 2.2, 13.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.


#### 5.2.2. Data Set II

Statistical methods are very helpful in estimating the random phenomena of wind speed. It is environment friendly and an alternative clean energy source in comparison with the fuels obtained from the fossil. It is a type of solar energy, which is determined by the Earth surface unequal heating. Wind speed probabilities are modelled with probability distribution as they are the important parameter of the wind power. Here, a data set is considered from the daily average wind speeds at Cairo city from [18] as follows: 3.5, 3.1, 3.8, 3.2, 3.2, 4.5, 5.6, 5.7, 4.9, 5.7, 4.3, 9.4, 9.3, 4.4, 2.7, 3.8, 4.9, 5.4, 4.9, 4.2, 5.4, 3.3, 6.9, 9.8, 10, 8, 5.6, 8.2, 9.4, 11.3, 9.4, 5.5, 4.9, 8.6, 5, 4.7, 3.8, 4.3, 6.7, 7.6, 13.3, 8.2, 5.8, 5.1, 7.8, 10.3, 9.3, 4.3, 7.4, 13.8, 10.7, 12, 8.9, 10.6, 6.8, 6.6, 11.1, 12.5, 14.4, 9.9, 4.8, 4.2, 5.5, 7.3, 12.4, 14.7, 6.4, 8.7, 5.2, 6.8, 5.6, 7.5, 7.7, 7.1, 6.1, 7.6, 5.8, 6.3, 12.2, 6, 3.5, 9.5, 8.8, 5.2, 5, 9.8, 8, 7.9, 6.8, 5.7, 7.3, 6.8, 4.7, 5.3, 9.6, 10.1, 7.3, 6.7, 5.4, 5.4.


## 5.1. Simulation Study

A simulation study is performed to obtain the average values of MLEs (maximum likelihood estimators), MSEs (mean square error), and bias. The following steps are used to perform the simulation.Step 1.
First, suppose the values of the parameters from KEGP asα,β,ξ,δ, that is, Case I = (2, 3, 2, 1), Case II = (3, 4, 2, 1), Case III = (1, 2, 4, 3), and Case IV = (1, 2, 2, 5).Step 2.
The process is repeated 10,000 times and the MSE and bias are computed for the estimates forn = 50, 100, 500, and 1000.Step 3.
Use the following quantile expression for the generation of random numbers from KEGP as(36)X=δ1−−log1−e−a1/1−e−α−u/αβ−ξ−1ξ,where U∼0,1.
Also, the bias and MSEs are calculated using the following expression:(37)MSE=1w∑i=0wbi^−b,bias=1w∑i=0wbi^−b2,where b=α,β,ξ,δ. Results of simulations were obtained for different combinations of the parameters α,β,ξ,andδ and are displayed in Table 1. It is exhibited clearly from the table that with the increase of sample size, these estimates are reasonably consistent and approaches the true values of parameters. Furthermore, the MSEs and bias decrease for all the combinations of parameters with increase of the sample size. Therefore, it has been determined that the process of MLE is performing well in estimating the parameters of KEGPD.Table 1
Average values of MLEs, MSEs, and bias.
Parametersnα^β^ξ^δ^MSE (α^)MSE (β^)MSE (ξ^)MSE (δ)Bias (α^)Bias (β^)Bias (ξ^)Bias (δ^)α = 2β = 3ξ=2δ=1502.82244.3792.38620.88919.74134.83441.73911.46040.82241.37900.3861−0.11081003.12283.6622.53941.317312.4443.059022.79103.18061.12270.66210.53940.317315002.29473.5612.16990.92981.95011.66310.41030.28840.29470.56080.1699−0.0701910002.17203.5672.09770.87340.83161.48080.15830.187220.17200.56710.0977−0.1265α = 3β = 4ξ=2δ=1504.04084.04852.51811.721923.36803.52192.90164.7391.04080.04850.51810.72191003.56144.59612.21831.422511.01024.47911.47472.30680.56140.59610.21830.42255002.76754.76891.93650.92591.74993.39650.21290.3039−0.23250.7689−0.0634−0.074110002.98274.62981.99920.94570.95942.6280.13120.2547−0.01730.6298−0.0007−0.0543α = 1β = 2ξ=4δ=3502.03672.5255.23044.239023.97112.885612.22017.4771.03670.52541.23041.23901002.29802.5495.35303.913011.14341.505512.891220.74281.29800.549241.35300.91305001.34292.3724.28493.41991.17441.06081.09587.52080.34300.37240.28490.4198910001.15632.3644.10103.00190.34200.91780.16504.08660.15630.36400.10100.0019α = 1β = 2ξ=2δ=5503.1713.26783.32614.514434.13795.760410.722419.58792.17111.26781.3261−0.48551002.5453.07133.2014.87629.0924.6516.118725.2251.54521.07131.2011−0.12435002.21872.5112.79854.80929.60561.49143.242010.34091.21870.51100.7985−0.190810001.68322.28122.4574.7914.41610.46601.55545.37460.68320.28120.4569−0.2091


## 5.2. Real-Life Applications

The practicality of the KEGP is demonstrated by using two real-life data sets. The first data set is considered from [17], and the second data set is taken from [18]. The KEGP is compared with various submodels such as Kumaraswamy Pareto [19], Alpha Power Pareto [20], Generalized Pareto [2], and Exponentiated Generalized Pareto [21] distributions with the following probability density functions:(i)
Kumaraswamy Pareto (KP) distribution:(38)fx=abkβkxk+11−βxka−11−1−βxkab−1,x≥β,a,b,k>0.(ii)
Alpha Power Pareto (APP):(39)fx=βlogαα−1α1−x−βx−β−1,α≠0,x>0,fx,α=0,x>0.(iii)
Generalized Pareto (GP):(40)fx=1δ1+ξxδ−1/ξ−1,ξ≠0,x≥0,δ>0.(iv)
Exponentiated Generalized Pareto (EGP):(41)fx=exδ1+ξexδ−1/ξ−1,ξ≠0,−∞<x<∞,δ>0.The performance of the KEGP is checked by comparing it with various forms of Pareto distributions discussed above using the goodness-of-fit test criteria. A package of R software, that is, Adequacy Model, is utilized which includes the outcome of Akaike’s Information Criteria (AIC), Bayesian Information Criterion (BIC), Consistent Akaike’s Information Criteria (CAIC), Hannan–Quinn Information Criteria (HQIC),−lnθ^, and Kolmogorov–Smirnov test (KS) and its p value. Generally, the model is considered as the best-fitted model to the data if the values of all the above criteria are smaller with a larger p value. Tables 2 and 3 display the MLEs and Kolmogorov–Smirnov (K-S) results along with their p values. It can be seen clearly that the results of the Khalil Extended Generalized Pareto distribution for the K-S test are smaller with a large p value, i.e., K-S value = 0.0342, p value = 0.9982 for data set I and K-S value = 0.0835, p value = 0.4881 for data set II, in comparison with its other variants, hence proving a better fit. Furthermore, Tables 4 and 5 display the results of AIC, BIC, CAIC, and HQIC for which the Khalil Extended Generalized Pareto distribution has lower values in comparison with its other various versions. A commendable performance of the KEGP distribution is also visible from Figures 4 and 5 displaying the empirical and theoretical pdfs and CDFs. Figures 6 and 7 demonstrate the Q-Q and P-P plots for both the data sets, respectively. The better performance of the suggested model indicates the need of introduction of new distributions in managing some sets of real data.Table 2
MLEs and standard errors (in parenthesis), K-S values, andp values of data set I.
ModelMLEs of the parameterK-Sp valuesKEGPDα,β,ξ,δ−3.8740, 0.7348, 0.30490, 3.5039 (2.111, 0.519, 0.0891, 0.950)0.03420.9982KP (β,a,b,k)0.05109, 0.3212, 7.1078, 4.3484 (0.009, 0.042, 1.047, 1.373)0.17540.00075APPα,β1.3800, 0.6023 (0.500, 0.0579)1.8082.2e − 16GPξ,δ8.8108, 0.0647 (1.039, 0.078)0.09410.2064EGPξ,δ20.4552, 8.3284 (6.464, 0.794)0.050020.9059Table 3
MLEs and standard errors (in parenthesis), K-S values, andp values of data set II.
ModelMLEs of the parameterK-Sp valuesKEGPDα,β,ξ,δ0.8836, 7.9088, −0.1279, 3.4898 (5.816, 20.854, 0.836, 9.016)0.08350.4881KP (β,a,b,k)1.5574, 0.8636, 7.7916, 9.7403 (0.936, 6.909, 0.179, 3.9484)0.09400.3324APPα,β36.1665, 1.0601 (10.175, 0.068)0.34121.537e − 10GPξ,δ11.6718, −0.7904 (0.583, 0.037)0.24779.291e − 06EGPξ,δ49.9512, 4.0312 (11.410, 0.520)0.26851.086e − 06Table 4
AIC, BIC, CAIC, HQIC, and -L for data set I.
ModelsAICCAICBICHQIC−LKEGPDα,β,ξ,δ830.0632830.3884841.4713834.6984411.0316KP (β,a,b,k)894.9182895.2434906.3263899.5533443.459APPα,β848.9063849.0023854.6103851.2238422.4531GPξ,δ833.7435833.8395839.4476836.0611414.8718EGPξ,δ842.0753842.1713847.7794844.3929419.0377Table 5
AIC, BIC, CAIC, HQIC, and −L for data set II.
ModelsAICCAICBICHQIC−LKEGPDα,β,ξ,δ478.3736478.7947488.7943482.5911235.1868KPβ,a,b,k480.2818480.7029490.7025484.4992236.6409APPα,β614.2206614.3443619.4309616.3293305.1103GPξ,δ537.4365537.5602542.6468539.5452266.7182EGPξ,δ527.2457527.3694532.456529.3544261.6228Figure 4
Density comparison of data set I.Figure 5
Density comparison of data set II.Figure 6
Quantiles and probability plots of data set I.Figure 7
Quantiles and probability plots for data set II.


### 5.2.1. Data Set I

The first data set is considered from [17], which consists of remission time (in months) of bladder cancer patients. The data set is positively skewed and unimodal, with a skewness of 3.286, mean remission time of 9.366 months, and standard deviation of 10.508 months. These data are recently studied in [22, 23] as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.2, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.4, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.5, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.7, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.26, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.09, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.4, 3.02, 4.43, 5.71, 7.93, 11.79, 18.10, 1.46, 4.4, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.5, 6.25, 8.37, 12.02, 2.2, 13.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.


### 5.2.2. Data Set II

Statistical methods are very helpful in estimating the random phenomena of wind speed. It is environment friendly and an alternative clean energy source in comparison with the fuels obtained from the fossil. It is a type of solar energy, which is determined by the Earth surface unequal heating. Wind speed probabilities are modelled with probability distribution as they are the important parameter of the wind power. Here, a data set is considered from the daily average wind speeds at Cairo city from [18] as follows: 3.5, 3.1, 3.8, 3.2, 3.2, 4.5, 5.6, 5.7, 4.9, 5.7, 4.3, 9.4, 9.3, 4.4, 2.7, 3.8, 4.9, 5.4, 4.9, 4.2, 5.4, 3.3, 6.9, 9.8, 10, 8, 5.6, 8.2, 9.4, 11.3, 9.4, 5.5, 4.9, 8.6, 5, 4.7, 3.8, 4.3, 6.7, 7.6, 13.3, 8.2, 5.8, 5.1, 7.8, 10.3, 9.3, 4.3, 7.4, 13.8, 10.7, 12, 8.9, 10.6, 6.8, 6.6, 11.1, 12.5, 14.4, 9.9, 4.8, 4.2, 5.5, 7.3, 12.4, 14.7, 6.4, 8.7, 5.2, 6.8, 5.6, 7.5, 7.7, 7.1, 6.1, 7.6, 5.8, 6.3, 12.2, 6, 3.5, 9.5, 8.8, 5.2, 5, 9.8, 8, 7.9, 6.8, 5.7, 7.3, 6.8, 4.7, 5.3, 9.6, 10.1, 7.3, 6.7, 5.4, 5.4.


## 5.2.1. Data Set I

The first data set is considered from [17], which consists of remission time (in months) of bladder cancer patients. The data set is positively skewed and unimodal, with a skewness of 3.286, mean remission time of 9.366 months, and standard deviation of 10.508 months. These data are recently studied in [22, 23] as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.2, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.4, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.5, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.7, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.26, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.09, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.4, 3.02, 4.43, 5.71, 7.93, 11.79, 18.10, 1.46, 4.4, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.5, 6.25, 8.37, 12.02, 2.2, 13.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.


## 5.2.2. Data Set II

Statistical methods are very helpful in estimating the random phenomena of wind speed. It is environment friendly and an alternative clean energy source in comparison with the fuels obtained from the fossil. It is a type of solar energy, which is determined by the Earth surface unequal heating. Wind speed probabilities are modelled with probability distribution as they are the important parameter of the wind power. Here, a data set is considered from the daily average wind speeds at Cairo city from [18] as follows: 3.5, 3.1, 3.8, 3.2, 3.2, 4.5, 5.6, 5.7, 4.9, 5.7, 4.3, 9.4, 9.3, 4.4, 2.7, 3.8, 4.9, 5.4, 4.9, 4.2, 5.4, 3.3, 6.9, 9.8, 10, 8, 5.6, 8.2, 9.4, 11.3, 9.4, 5.5, 4.9, 8.6, 5, 4.7, 3.8, 4.3, 6.7, 7.6, 13.3, 8.2, 5.8, 5.1, 7.8, 10.3, 9.3, 4.3, 7.4, 13.8, 10.7, 12, 8.9, 10.6, 6.8, 6.6, 11.1, 12.5, 14.4, 9.9, 4.8, 4.2, 5.5, 7.3, 12.4, 14.7, 6.4, 8.7, 5.2, 6.8, 5.6, 7.5, 7.7, 7.1, 6.1, 7.6, 5.8, 6.3, 12.2, 6, 3.5, 9.5, 8.8, 5.2, 5, 9.8, 8, 7.9, 6.8, 5.7, 7.3, 6.8, 4.7, 5.3, 9.6, 10.1, 7.3, 6.7, 5.4, 5.4.


## 6. Conclusion

A new and improved generalization of the Generalized Pareto distribution is proposed in this paper named as Khalil Extended Generalized Pareto distribution (KEGP). The new proposed generalization (KEGP) of the Generalized Pareto proved to be more flexible and suitable for monotone as well as nonmonotone life time data. In addition, it is observed that the hazard function of the new proposed KEGP model is more flexible in fitting monotonically increasing, decreasing, and various types of data. For the proposed model, various statistical properties are derived. The estimation of parameters is done using the famous method of maximum likelihood. Furthermore, the consistency of the parameters is proved using the Monte Carlo simulation method. Moreover, the practicality of the distribution is exemplified with the help of two real-life data sets. Finally, among the fitted models, the KEGP provided a better fit than its other submodels.



---

*Source: 6855652-2021-12-27.xml*