# A New Class of the Power Function Distribution: Theory and Inference with an Application to Engineering Data

Authors: Alya Al Mutairi; Muhammad Z. Iqbal; Muhammad Z. Arshad; Ahmed Z. Afify
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, a new class that generates optimal univariate models called a new exponentiated-G class of distributions is developed. Numerous complementary statistical properties are derived and discussed in detail for the newly exponentiated power function (EPF) distribution. All possible shapes of the probability density and hazard rate functions are sketched for selected values of parameters. Six accredited estimation methods are discussed, and their performance is assessed and compared by a simulation study. The applicability of the new class is evaluated by analyzing the automotive engineering sector data.


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## Body

## 1. Introduction

Modeling complicated problems is an enigma for applied researchers and practitioners. They seem to be worried about dealing with a variety of lifetime datasets that particularly follows physical and natural sciences. For this, they are searching for simple and efficient models. Consequently, a power function (PF) distribution is explored. It is a simple lifetime model as exponential and Pareto distributions. The PF distribution is a special case of the beta distribution. For more details about the classical work of the PF distribution, see Dallas [1] who developed a relationship between the Pareto and PF variables through an inverse transformation. Furthermore, for a deep understanding of the characterization of the PF distribution, see some credible work of Meniconi and Barry [2], Saran and Pandey [3], Chang [4], Tavangar [5], and Ahsanullah et al. [6].In the most recent times, attention towards the generalization of probability distributions has grown phenomenally high. For more insight, see the trustworthy work of Cordeiro and Brito [7], Zaka and Akhter [8], Al Mutairi et al. [9], Tahir et al. [10], Shahzad et al. [11], Ahsan-ul-Haq et al. [12], Okorie et al. [13], Abdul-Moniem [14], Hassan et al. [15], Zaka et al. [16], Arshad et al. [17], Arshad et al. [18, 19], Al-Mutairi [20], Alzaatreh et al. [21], Gleaton and Lynch [22], Bourguignon et al. [23], Afify et al. [24], Tahir et al. [25], Aldahlan et al. [26], Aslam et al. [27], Balogun et al. [28], Afify et al. [29], Mansour et al. [30], Mahdavi and Kundu [31], Nassar et al. [32], Ijaz et al. [33], Klakattawi and Aljuhani [34], Afify et al. [35], Alsubie et al. [36], Ahmad et al. [37], and Nofal et al. [38].To the best of our knowledge, the new class has not been discussed before, and it is the first time to explore the scenario particularly observed in the automobile sector. We develop a new class of distributions called the new exponentiated-G (NE-G) family and study one of its special submodels using the PF distributions as a baseline model. The studied model is called the exponentiated power function (EPF) distribution. The present study has some motivations as follows: (a) to develop new optimal models; (b) to advance the characteristics of the baseline models; (c) the density and hazard rate functions possess unimodal and bathtub-shaped curves, respectively; (d) to model the real-time scenario in the automobile sector.This paper is outlined in the following sections. The development of the new class is presented in Section2. A comprehensive discussion on mathematical and reliability measures is completed in Sections 3 and 4, respectively. Miscellaneous measures are discussed in Section 5. Six accredited estimation methods are discussed in Section 6. Simulation results are presented in Section 7. A lifetime application of the EPF distribution is discussed in Section 8, and finally, some conclusions are reported in Section 9.


## 2. The New Exponentiated-G Class

Tahir and Cordeiro [39] (Remarks 2 (ii)) developed the exponentiated generalized negative binomial G class which is defined by the CDF:(1)Fx;ℶ=1−1+ηγGx;ℶβ−1/α1−1+ηγ−1/α.Chesneau et al. [40] reparameterized the parameters of (1) and provided the following CDF:(2)Fx;ℶ=1+ηGx;ℶ1+1/η−11+η1+1/η−1.In this section, we provide a new generator of distributions that is very simple and capable of generating optimal alternative models. The new generator is called the new exponentiated-G class, and it is specified by the CDF:(3)Fx;ℶ=1+Gx;ℶα−12α−1,x∈ℜ,where Gx;ℶϵ0,1 is a CDF of any baseline (parent) distribution. It is based on a parametric vector ℶ>0 depending on (r× 1), α>0 is a shape parameter, and 1/2α−1 is a normalizing constant. F0;ℶ=0, and F∞;ℶ=1. It is noted that the new class reduces to the baseline model with α=1.The probability density function (PDF) (f(x)) and the hazard rate function (HRF) (h(x)) of the new class reduce to(4)fx;ℶ=αgx;ℶ1+Gx;ℶα−12α−1,x∈ℜ,(5)hx;ℶ=αgx;ℶ1+Gx;ℶα−12α−1+Gx;ℶα−1,x∈ℜ,where gx;ℶ=dGx;ℶ/dx.The quantile function (QF) (Q(x)) of the new class takes the form.(6)Qx=G−11+2α−1p1α−1,p∈0,1.


### 2.1. EPF Distribution

In this section, we discuss some useful characteristics of the EPF distribution. The PF distribution is specified by the following CDF and PDF:(7)Gx=xg0β,β>0,0<x≤g0,and(8)gx=βg0βxβ−1,β>0,0<x≤g0.To this end, we define the analytical expressions of CDF and PDF of the new EPF distribution with two shape parametersα and β. The CDF of the EPF distribution has the form (for 0<x≤g0)(9)Fx=1+x/g0βα−12α−1,α,β>0.Its PDF reduces to(10)fx=αβg0β2α−11x1−β1+xg0β−1−α,α,β>0,where 1/2α−1 is the normalizing constant and g0 is a possible maximum assured life of a component.The EPF distribution brings an additional shape parameter (α>0) that modulates the skewness and kurtosis tail weights of the baseline model. We note that a new parameter may offer a better fit to the unimodal, increasing, U-shaped, and bathtub-shaped failure rate data. The EPF distribution reduces to the baseline model (power function) for α  = 1.


### 2.2. Asymptotic Properties of PDF and CDF

Asymptotes of the CDF and PDF atx⟶0 are given by(11)Fx|x⟶0∼0andfx|x⟶0∼0.Asymptotes of the CDF and PDF atx⟶g are given by(12)Fx|x⟶g0∼1andfx|x⟶g0∼αβ2α−1g02α−1.The derived expressions explore a dynamic effect ofα on the asymptotes of Fx and fx.


### 2.3. Shapes of PDF

Here, we discuss different shapes of the PDF of the EPF distribution. Figure1 presents some different curves of the PDF for various choices of EPF parameters. We note that these curves can be decreasing, decreasing-increasing, and can be upside-down bathtub for g0 = 2.Figure 1
Some curves of PDF for EPF distribution.


## 2.1. EPF Distribution

In this section, we discuss some useful characteristics of the EPF distribution. The PF distribution is specified by the following CDF and PDF:(7)Gx=xg0β,β>0,0<x≤g0,and(8)gx=βg0βxβ−1,β>0,0<x≤g0.To this end, we define the analytical expressions of CDF and PDF of the new EPF distribution with two shape parametersα and β. The CDF of the EPF distribution has the form (for 0<x≤g0)(9)Fx=1+x/g0βα−12α−1,α,β>0.Its PDF reduces to(10)fx=αβg0β2α−11x1−β1+xg0β−1−α,α,β>0,where 1/2α−1 is the normalizing constant and g0 is a possible maximum assured life of a component.The EPF distribution brings an additional shape parameter (α>0) that modulates the skewness and kurtosis tail weights of the baseline model. We note that a new parameter may offer a better fit to the unimodal, increasing, U-shaped, and bathtub-shaped failure rate data. The EPF distribution reduces to the baseline model (power function) for α  = 1.


## 2.2. Asymptotic Properties of PDF and CDF

Asymptotes of the CDF and PDF atx⟶0 are given by(11)Fx|x⟶0∼0andfx|x⟶0∼0.Asymptotes of the CDF and PDF atx⟶g are given by(12)Fx|x⟶g0∼1andfx|x⟶g0∼αβ2α−1g02α−1.The derived expressions explore a dynamic effect ofα on the asymptotes of Fx and fx.


## 2.3. Shapes of PDF

Here, we discuss different shapes of the PDF of the EPF distribution. Figure1 presents some different curves of the PDF for various choices of EPF parameters. We note that these curves can be decreasing, decreasing-increasing, and can be upside-down bathtub for g0 = 2.Figure 1
Some curves of PDF for EPF distribution.


## 3. Moments and Related Measures

Theorem 1.
LetX follow the EPF distribution with two shape parameters α,β>0; then, the r-th ordinary moment (μr') of X has the form(13)μr'=αβg0β2α−1∑j=0∞ψjg0Λr,j,βΛr,j,β.Proof.
The following expression follows using (6) as(14)μr'=αβg0β2α−1∫0g0xrxβ−11+xg0βα−1dx.
The prior expression can be rewritten by simplifying the expression1+x/g0βα−1 as follows:(15)μr'=αβg0β2α−1∑j=0∞α−1j1g0βj+1∫0g0xr+βj+1−1dx.
After some algebra, ther-th ordinary moment of X reduces to(16)μr'=αβg0β2α−1∑j=0∞ψjg0Λr,j,βΛr,j,β,where ψj=α−1j1/g0βj+1,Λr,j,β=r+βj+1.
Table1 presents some numerical results for the first-four ordinary moments (μ1′,μ2′,μ3′,μ4′), variance =  σ2, skewness =  γ1, and kurtosis =  γ2 for different values of the EPF parameters.
Table1 shows flexible and versatile behavior for moments, variance and alongside the skewness and kurtosis. The results indicate that the EPF distribution can be discussed for leptokurtic and skewed datasets.Table 1
Numerical analysis for moments, variance, skewness, and kurtosis.
Parametersμ′1μ′2μ′3μ′4σ2γ1γ2g0 = 2α = 1.5β = 1.91.35862.06213.33155.59330.32450.00110.0340β = 0.91.00441.35032.03703.27230.66620.00010.2197β = 0.10.30960.49151.00252.27830.45916.05267.7215β = 0.20.55860.92901.92864.42350.82092.40663.4680β = 0.30.76381.32152.78726.44781.11231.27302.0790β = 0.3α = 0.50.62261.03342.14154.90610.89901.98722.9723g0 = 3α = 0.10.56870.92721.90704.35110.81642.36763.4399β = 0.3α = 1.10.70641.20292.51965.80651.02651.52332.3952β = 0.09α = 1.10.22820.35610.72211.63580.33879.064711.1686β = 0.08α = 0.010.16670.25530.51411.16080.246013.367616.0774Corollary 1.
The first and second ordinary moments and the inverse moment (μ−w') can be obtained by substituting r=1,2, and –w, in (7), respectively. The analytical expressions of mean, variance, and inverse moment are given, respectively, by(17)EX=μ1'=αβg0β2α−1∑j=0∞ψjg0Λ1,j,βΛ1,j,β,VarX=αβg0β2α−1∑j=0∞ψjg0Λ2,j,βΛ2,j,β−αβg0β2α−1∑j=0∞ψjg0Λ1,j,βΛ1,j,β,
and(18)μ−w/=αβg0β2α−1∑j=0∞ψjg0Λ−w,j,βΛ−w,j,β.Corollary 2.
The factorial generating function is obtained directly followed byFxs=E1+sx=Eexln1+s=∑r=0∞ln1+sr/r!μr′, and it can be written for X as follows:(19)Fxs=αβg0β2α−1∑r=0∞ln1+srr!∑j=0∞ψjg0Λr,j,βΛr,j,β.Theorem 2.
IfX ∼ EPF α,β, then the moment generating function (MGF) (MXs) of X is given by(20)MXs=αβg0β2α−1∑r=0∞srr!∑j=0∞ψjg0Λr,j,βΛr,j,β,r=1,2,…..Proof.
The MGFMXs is defined as(21)MXs=∫0g0esxfxdx.
Also,esx=∑r=0∞sxr/r!.
Hence, the MGF ofX is obtained as(22)MXs=αβg0β2α−1∑r=0∞srr!∑j=0∞ψjg0Λr,j,βΛr,j,β.Theorem 3.
.
IfX ∼ EPF α,β, then the characteristic function (φXs) of X is given by(23)φXs=αβg0β2α−1∑r=0∞isrr!∑j=0∞ψjg0Λr,j,βΛr,j,β;r=1,2,…andi=−1.Proof.
The characteristic functionφXs is defined as(24)φXs=∫0g0eisxfxdx.
Hence, the characteristic function ofX is obtained as(25)φXs=αβg0β2α−1∑r=0∞isrr!∑j=0∞ψjg0Λr,j,βΛr,j,β.
Vitality function is defined as(26)Vx=1Sx∫xg0xfxdx.
It is obtained forX as(27)Vx=11−Fxαβg0β2α−1∑j=0∞ψjg0Λ1,j,β−xΛ1,j,βΛ1,j,β.
The conditional moments are defined as(28)Exr|X>t=1F¯t∫tg0xrfxdx.
It is obtained forX as(29)Exr|X>t=11−Ftαβg0β2α−1∑j=0∞ψjg0Λr,j,β−tΛr,j,βΛr,j,β.


### 3.1. Incomplete Moments and Associated Measures

Theorem 4.
IfX ∼ EPF α,β, then the r-th lower incomplete moments of X are(30)Φrt=αβg0β2α−1∑j=0∞ψjtΛr,j,βΛr,j,β.Proof.
Ther-th incomplete moments Φrt are defined as(31)Φrt=∫0txrfxdx.
It is obtained forX as(32)Φrt=αβg0β2α−1∑j=0∞α−1j1g0βj+1∫0txr+βj+1−1dx.
Hence,Φrt of X reduces to(33)Φrt=αβg0β2α−1∑j=0∞ψjtΛr,j,βΛr,j,β.Corollary 3.
The first incomplete moment is obtained by substitutingr = 1 in equation (33) as(34)Φ1t=αβg0β2α−1∑j=0∞ψjtΛ1,j,βΛ1,j,β.
The residual life function is defined as(35)Rtx=Sx+tSt.
Hence, the residual life function (RLF) and its associated CDF ofX are given by(36)Rtx=2α−1+x+t/gβα2α−1+t/g0βα,
and(37)FRt=1+x+t/gβα−1+t/gβα2α−1+t/g0βα,respectively.
Furthermore, the reversed RLF is defined asR¯tx=Sx−t/St. Hence, reversed RLF and its associated CDF of X take the forms:(38)R¯tx=2α−1+x−t/g0βα2α−1+t/g0βα,FR¯t=1+x−t/g0βα−1+t/g0βα2α−1+t/g0βα.
The mean RLF is defined asMRL=1−Φ1t/St−t. It is obtained for X as(39)MRL=1St−t1−αβg0β2α−1∑j=0∞ψjtΛ1,j,βΛ1,j,β.
The mean inactivity time (MIT) is defined asMIT=t−Φ1t/Ft. It is obtained for X as(40)MIT=t−αβ∑j=0∞ψjtΛ1,j,β/Λ1,j,βg0β1+x/g0βα−1.
The strong mean inactivity time (SMIT) of a device is defined asSMIT=t2−Φ2t/Ft. It is obtained for X as(41)MIT=t2−αβ∑j=0∞ψjtΛ1,j,β/Λ1,j,βg0β1+x/g0βα−1.
The mean past lifetime (MPL) of a device is defined as MPL =x−∫0xtftdt/Fx. It is obtained for X as(42)MPL=x1+x/g0βα−1−2α−1αβ/g0β2α−1∑j=0∞ψjxΛ1,j,β/Λ1,j,β1+x/g0βα−1.
Furthermore, the Lorenzt, Bonferroni Bt, and Zenga Zt inequality curves have a significant role not only in the study of economics, the distribution of income, poverty, or wealth, but also they have a vital role in the fields of insurance, demography, medicine, and reliability engineering.Theorem 5.
IfX∼ EPFα,β, then the Lorenz inequality curve of X is(43)Lt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β.Proof.
Lorenz inequality curve is defined as(44)Lt=Φ1tμ1′.
It is obtained forX, using equations (17) and (34), as(45)Lt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β.Theorem 6.
IfX∼ EPFα,β, then the Bonferroni inequality curve of X is as follows:(46)Bt=2α−1∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β1+x/g0βα−1.Proof.
The Bonferroni inequality curve is defined as(47)Bx=LtFx.
It is obtained forX, using Lt and the CDF of the EPF model, as(48)Bt=2α−1∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β1+x/g0βα−1.Theorem 7.
IfX∼ EPFα,β, then the Zenga inequality curve of X reduces to(49)Zt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β−t∑j=0∞ψjg0Λ1,j,β/Λ1,j,βt∑j=0∞ψjg0Λ1,j,β/Λ1,j,β−∑j=0∞ψjtΛ1,j,β/Λ1,j,β.Proof.
Zenga inequality curve is defined as(50)Zt=Lt−tt1−Lt.
It is obtained forX as(51)Zt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β−t∑j=0∞ψjg0Λ1,j,β/Λ1,j,βt∑j=0∞ψjg0Λ1,j,β/Λ1,j,β−∑j=0∞ψjtΛ1,j,β/Λ1,j,β.


## 3.1. Incomplete Moments and Associated Measures

Theorem 4.
IfX ∼ EPF α,β, then the r-th lower incomplete moments of X are(30)Φrt=αβg0β2α−1∑j=0∞ψjtΛr,j,βΛr,j,β.Proof.
Ther-th incomplete moments Φrt are defined as(31)Φrt=∫0txrfxdx.
It is obtained forX as(32)Φrt=αβg0β2α−1∑j=0∞α−1j1g0βj+1∫0txr+βj+1−1dx.
Hence,Φrt of X reduces to(33)Φrt=αβg0β2α−1∑j=0∞ψjtΛr,j,βΛr,j,β.Corollary 3.
The first incomplete moment is obtained by substitutingr = 1 in equation (33) as(34)Φ1t=αβg0β2α−1∑j=0∞ψjtΛ1,j,βΛ1,j,β.
The residual life function is defined as(35)Rtx=Sx+tSt.
Hence, the residual life function (RLF) and its associated CDF ofX are given by(36)Rtx=2α−1+x+t/gβα2α−1+t/g0βα,
and(37)FRt=1+x+t/gβα−1+t/gβα2α−1+t/g0βα,respectively.
Furthermore, the reversed RLF is defined asR¯tx=Sx−t/St. Hence, reversed RLF and its associated CDF of X take the forms:(38)R¯tx=2α−1+x−t/g0βα2α−1+t/g0βα,FR¯t=1+x−t/g0βα−1+t/g0βα2α−1+t/g0βα.
The mean RLF is defined asMRL=1−Φ1t/St−t. It is obtained for X as(39)MRL=1St−t1−αβg0β2α−1∑j=0∞ψjtΛ1,j,βΛ1,j,β.
The mean inactivity time (MIT) is defined asMIT=t−Φ1t/Ft. It is obtained for X as(40)MIT=t−αβ∑j=0∞ψjtΛ1,j,β/Λ1,j,βg0β1+x/g0βα−1.
The strong mean inactivity time (SMIT) of a device is defined asSMIT=t2−Φ2t/Ft. It is obtained for X as(41)MIT=t2−αβ∑j=0∞ψjtΛ1,j,β/Λ1,j,βg0β1+x/g0βα−1.
The mean past lifetime (MPL) of a device is defined as MPL =x−∫0xtftdt/Fx. It is obtained for X as(42)MPL=x1+x/g0βα−1−2α−1αβ/g0β2α−1∑j=0∞ψjxΛ1,j,β/Λ1,j,β1+x/g0βα−1.
Furthermore, the Lorenzt, Bonferroni Bt, and Zenga Zt inequality curves have a significant role not only in the study of economics, the distribution of income, poverty, or wealth, but also they have a vital role in the fields of insurance, demography, medicine, and reliability engineering.Theorem 5.
IfX∼ EPFα,β, then the Lorenz inequality curve of X is(43)Lt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β.Proof.
Lorenz inequality curve is defined as(44)Lt=Φ1tμ1′.
It is obtained forX, using equations (17) and (34), as(45)Lt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β.Theorem 6.
IfX∼ EPFα,β, then the Bonferroni inequality curve of X is as follows:(46)Bt=2α−1∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β1+x/g0βα−1.Proof.
The Bonferroni inequality curve is defined as(47)Bx=LtFx.
It is obtained forX, using Lt and the CDF of the EPF model, as(48)Bt=2α−1∑j=0∞ψjtΛ1,j,β/Λ1,j,β∑j=0∞ψjg0Λ1,j,β/Λ1,j,β1+x/g0βα−1.Theorem 7.
IfX∼ EPFα,β, then the Zenga inequality curve of X reduces to(49)Zt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β−t∑j=0∞ψjg0Λ1,j,β/Λ1,j,βt∑j=0∞ψjg0Λ1,j,β/Λ1,j,β−∑j=0∞ψjtΛ1,j,β/Λ1,j,β.Proof.
Zenga inequality curve is defined as(50)Zt=Lt−tt1−Lt.
It is obtained forX as(51)Zt=∑j=0∞ψjtΛ1,j,β/Λ1,j,β−t∑j=0∞ψjg0Λ1,j,β/Λ1,j,βt∑j=0∞ψjg0Λ1,j,β/Λ1,j,β−∑j=0∞ψjtΛ1,j,β/Λ1,j,β.


## 4. Reliability Function and Associated Measures

Probability distributions consider as a backbone for reliability engineering to analyze and predict the lifetime of a component/device. In this section, numerous notable reliability measures are discussed.


### 4.1. Survival Function

The survival function ofX takes the form:(52)Sx=2α−1+x/g0βα2α−1.


### 4.2. Hazard Rate Function

The HRF (in demography), failure rate function (in engineering), and sometimes it is called the force of mortality (in economics). The HRF ofX is(53)hx=αβxβ−11+x/g0βα−1g0β2α−1+x/g0βα.


### 4.3. Mean Time between Failures

Mean time between failures (MTBF) is defined as−t/logSx Hence, it is obtained for X as(54)MTBF=−tlog2α−1−12α−1+x/g0βα.


### 4.4. Cumulative HRF

The cumulative HRF is defined ashcx=−logSx. Hence, the cumulative HRF of X has the form:(55)hcx=−log2α−1+x/g0βα2α−1.Figure2 presents the different curves of the EPF of HRF for various choices of its parameters. We note that it possesses increasing U-shaped and bathtub shape curves for g0 = 3.Figure 2
Some curves of the HRF for the EPF distribution.


### 4.5. Reverse HRF

The reverse HRF is defined ashrx=fx/Sx. It is obtained for X as(56)hrx=αβxβ−11+x/g0βα−1g0β2α−1+x/g0βα.


### 4.6. Odds Ratio

The odds ratio is defined asOx=Fx/fx. Hence, the odds ratio of X is given by(57)Ox=g0β1+x/g0βα−1αβxβ−11+x/g0βα−1.


### 4.7. Mills Ratio

The mill’s ratio is defined asMx=Sx/fx. Hence, the mill’s ratio of X is(58)Mx=g0β2α−1+x/g0βααβxβ−11+x/g0βα−1.


## 4.1. Survival Function

The survival function ofX takes the form:(52)Sx=2α−1+x/g0βα2α−1.


## 4.2. Hazard Rate Function

The HRF (in demography), failure rate function (in engineering), and sometimes it is called the force of mortality (in economics). The HRF ofX is(53)hx=αβxβ−11+x/g0βα−1g0β2α−1+x/g0βα.


## 4.3. Mean Time between Failures

Mean time between failures (MTBF) is defined as−t/logSx Hence, it is obtained for X as(54)MTBF=−tlog2α−1−12α−1+x/g0βα.


## 4.4. Cumulative HRF

The cumulative HRF is defined ashcx=−logSx. Hence, the cumulative HRF of X has the form:(55)hcx=−log2α−1+x/g0βα2α−1.Figure2 presents the different curves of the EPF of HRF for various choices of its parameters. We note that it possesses increasing U-shaped and bathtub shape curves for g0 = 3.Figure 2
Some curves of the HRF for the EPF distribution.


## 4.5. Reverse HRF

The reverse HRF is defined ashrx=fx/Sx. It is obtained for X as(56)hrx=αβxβ−11+x/g0βα−1g0β2α−1+x/g0βα.


## 4.6. Odds Ratio

The odds ratio is defined asOx=Fx/fx. Hence, the odds ratio of X is given by(57)Ox=g0β1+x/g0βα−1αβxβ−11+x/g0βα−1.


## 4.7. Mills Ratio

The mill’s ratio is defined asMx=Sx/fx. Hence, the mill’s ratio of X is(58)Mx=g0β2α−1+x/g0βααβxβ−11+x/g0βα−1.


## 5. Miscellaneous Measures

### 5.1. Quantile Function

Theorem 8.
IfX∼ EPFα,β, then the QF of X is given by(59)QF=g02α−1p+11/α−11/β.Proof.
The QF is defined by(60)QF=F−1xp=PX≤xp,p∈0,1.
Thep-th QF of X is obtained, by inverting the CDF (7), as(61)QF=g02α−1p+11/α−11/β.Corollary 4.
The 1st quartile (Q1), median (Q2), and 3rd quartile (Q3) of X are obtained by substituting p = 0.25, 0.5, and 0.75 in (61), respectively. The analytical expressions are(62)QFQ1=g02α−10.25+11/α−11/β,QFQ2=g02α−10.50+11/α−11/β,and(63)QFQ3=g02α−10.75+11/α−11/β,respectively.


### 5.2. Skewness and Kurtosis

Bowley’s [41] and Moors’s [42] coefficients of skewness and kurtosis can be calculated by the following two equations:(64)Bsk=Q0.75+Q0.25−2Q0.50Q0.75−Q0.25,Mkr=Q0.375−Q0.125−Q0.625+Q0.875Q0.75−Q0.25.Quartiles and octiles based on these descriptive measures provide more robust estimates than the traditional skewness and kurtosis measures. We note that these measures are almost less reactive to outliers and work more effectively for the distributions, deficient in moments. Figure3 illustrates the skewness and kurtosis curves for the EPF distribution. We note that skewness and kurtosis are expressed as a function of β. Figure 3 illustrates a positive to negative trend of skewness, and an increasing trend in kurtosis can be observed with the increase of α.Figure 3
Skewness (left panel) and Kurtosis (right panel) curves for EPF distribution is(65)IδX=11−δlog∫0gofδxdx,δ>0andδ≠1.


### 5.3. Entropy Measures

Kurtosis and entropy measures have the same role in comparing the shapes and tail weights of various density functions. The entropy of a random variableX is defined as a measure of uncertainty.In this section, we have developed numerous well-known entropy measures including Rényi [43], Havrda and Charvat [44], and Mathai and Haubold [45]. For more details, see some promising work of Basit et al. [46], Dey et al. [47], and Ijaz et al. [48].Theorem 9.
IfX∼ EPFα,β, then the Rényi entropy of X is(66)11−δlogαβ2α−1δ∑i=0∞δα−1ig0−βi+1goβi+δβ−1+1βi+δβ−1+1;δ>0andδ≠1.Proof.
The Rényi entropy forX is defined by
Using Equation (10), we can write(67)fδx=αβg0β2α−1δxδβ−11+xg0βδα−1.
Then, integrationfδx gives(68)∫0gofδxdx=αβ2α−1δ∑i=0∞δα−1igo−βi+1goβi+δβ−1+1βi+δβ−1+1.
Hence, the Rényi entropy reduces to(69)IδX=11−δlogαβ2α−1δ∑i=0∞δα−1ig0−βi+1goβi+δβ−1+1βi+δβ−1+1.
The expression developed in (68) is quite helpful in the further computation of entropy measures of Havrda and Charvat, and Mathai and Haubold. The final expressions of Havrda and Charvat, and Mathai and Haubold entropy measures are presented in Table 2.
Table3 presents the results of Rényi, Havrda and Charvat, and Mathai and Haubold entropy measures for some choices of model parameters for (go  = 3), Set-I (α=1.1,β=2.1), Set -II (α=2.1,β=2.5), Set –III (α=0.1,β=0.6), and Set-IV (α=0.01,β=1.1).
A wide range of positive and negative values of entropy measures makes the EPF distribution more flexible and versatile.Table 2
The final expressions for Havrda and Charvat, and Mathai and Haubold entropy measures.
Entropy/SupportMeasureExpressionHavrda and Charvatω>0,ω≠11/1−ω∫0gofωxdx−111−ωαβ2α−1ω∑i=0∞ωα−1ig0−βi+1×goβi+ωβ−1+1βi+ωβ−1+1−1Mathai and Hauboldϑ>0,ϑ≠11/ϑ−1∫0gof2−ϑxdx−11ϑ−1αβ2α−12−ϑ∑i=0∞2−ϑα−1ig0−βi+1×goβi+2−ϑβ−1+1βi+2−ϑβ−1+1−1Table 3
Numerical analysis for Rényi, Havrda and Charvat, and Mathai and Haubold entropy measures.
EntropyInt.Set -ISet -IISet -IIISet-IVRényiδ=1.148.507345.556842.999936.4934δ=1.513.229212.424611.72729.9527δ=1.710.709410.05809.49348.0570δ=1.99.30948.74328.25257.0038Havrda and Charvatω=1.120.821420.913020.745021.0333ω=1.54.66574.74724.49614.8391ω=1.73.46373.53893.23073.6180ω=1.92.77852.84732.45102.9149Mathai and Hauboldℶ=1.1−19.0736−18.9800−19.1095−18.8459ℶ=1.5−2.7702−2.6883−2.7356−2.5421ℶ=1.7−1.3975−1.3360−1.3444−1.2080ℶ=1.9−0.4388−0.4138−0.4027−0.3487


### 5.4. Distribution of Order Statistics

LetX1,…,Xn be a random sample of size n and their corresponding order statistics (OS) X1<...<Xn from the EPF distribution. The PDF of the i-th OS is fi:nx=1/Bi,n−i+1!Fxi−11−Fxn−ifx,i = 1, 2, 3, …, n.Thei-th OS density is obtained by incorporating Equations (6) and (8) in the last equation.(70)fi:nx=1Bi,n−i+1!1+x/g0βα−12α−1i−12α−1+x/g0βα2α−1n−iαβg0β2α−1xβ−11+x/g0βα−1.The minimum and maximum OS densities are obtained, respectively, by substitutingi=1,n in (70).Thei-th OS CDF is defined by(71)Gx=∑r=innr1−Fxn−rFrx.Thei-th OS CDF of the EPF distribution reduces to(72)Gx=∑r=1nnr1+x/g0βα−12α−1r1+x/g0βα−12α−1n−r.The mediani-th OS PDF is(73)fm+1:nx=2m+1!m!2fxFxm1−Fxm.TheXm+1 median OS PDF of the EPF distribution has the form:(74)fm+1:nx=2m+1!m!2αβg0β2α−1xβ−11+xg0βα−11+x/g0βα−12α−1m2α−1+x/g0βα2α−1m.Thei-th and j-th OS joint distribution is defined by(75)fxi,xj=CFxii−1Fxj−Fxij−i−11−Fxjn−jfxifxj.For thei-th and j-th OS joint distribution of the EPF model is as follows:(76)fxi,xj=Cαβ2xixjβ−1g02β2α−121+xig0β1+xjg0βα−11+x/g0βα−12α−1i−11−1+x/g0βα−12α−1n−j1+xj/g0βα−1+xi/g0βα2α−1j−i−1.


### 5.5. Bivariate and Multivariate Extensions

In this section, we develop the bivariate and multivariate extensions for the EPF distribution by following the Morgenstern family and the Clayton familyThe CDF of the bivariate EPF distribution followed by the Clayton family for the random vectorX,Y is(77)FV1,V2=1+η1−Z1v11−Z2v2Z1v1Z2v2,whereη≤1, Z1v1=1+x1/g0β1α1−1/2α1−1 and Z2v2=1+x2/g0β2α2−1/2α2−1.The CDF of the bivariate EPF distribution followed by the Morgenstern family for the random vectorV1,V2 is defined as(78)Cx,y=x−τ1+τ2+y−τ1+τ2−1−1/τ1+τ2;τ1+τ2≥0.Let v1∼ EPF (α1,β1), and v2∼ EPF (α2,β2). Then, we set x=Z1v1=1+x1/g0β1α1−1/2α1−1, and y=Z2v2=1+x2/g0β2α2−1/2α2−1.The CDF of the bivariate EPF distribution followed by the Clayton family for the random vectorV1,V2 is(79)Gv1,v2=1+x1/g0β1α1−12α1−1τ1+τ2+1+x2/g0β2α2−12α2−1τ1+τ2−1−1/τ1+τ2.A simplen-dimensional extension of the last version for EPF distribution has the form:(80)Hx1,x2,x3,…,xn=∑j=1n11+x1/g0βjαj−12αj−1τ1+τ2+1−n−1/τ1+τ2.


## 5.1. Quantile Function

Theorem 8.
IfX∼ EPFα,β, then the QF of X is given by(59)QF=g02α−1p+11/α−11/β.Proof.
The QF is defined by(60)QF=F−1xp=PX≤xp,p∈0,1.
Thep-th QF of X is obtained, by inverting the CDF (7), as(61)QF=g02α−1p+11/α−11/β.Corollary 4.
The 1st quartile (Q1), median (Q2), and 3rd quartile (Q3) of X are obtained by substituting p = 0.25, 0.5, and 0.75 in (61), respectively. The analytical expressions are(62)QFQ1=g02α−10.25+11/α−11/β,QFQ2=g02α−10.50+11/α−11/β,and(63)QFQ3=g02α−10.75+11/α−11/β,respectively.


## 5.2. Skewness and Kurtosis

Bowley’s [41] and Moors’s [42] coefficients of skewness and kurtosis can be calculated by the following two equations:(64)Bsk=Q0.75+Q0.25−2Q0.50Q0.75−Q0.25,Mkr=Q0.375−Q0.125−Q0.625+Q0.875Q0.75−Q0.25.Quartiles and octiles based on these descriptive measures provide more robust estimates than the traditional skewness and kurtosis measures. We note that these measures are almost less reactive to outliers and work more effectively for the distributions, deficient in moments. Figure3 illustrates the skewness and kurtosis curves for the EPF distribution. We note that skewness and kurtosis are expressed as a function of β. Figure 3 illustrates a positive to negative trend of skewness, and an increasing trend in kurtosis can be observed with the increase of α.Figure 3
Skewness (left panel) and Kurtosis (right panel) curves for EPF distribution is(65)IδX=11−δlog∫0gofδxdx,δ>0andδ≠1.


## 5.3. Entropy Measures

Kurtosis and entropy measures have the same role in comparing the shapes and tail weights of various density functions. The entropy of a random variableX is defined as a measure of uncertainty.In this section, we have developed numerous well-known entropy measures including Rényi [43], Havrda and Charvat [44], and Mathai and Haubold [45]. For more details, see some promising work of Basit et al. [46], Dey et al. [47], and Ijaz et al. [48].Theorem 9.
IfX∼ EPFα,β, then the Rényi entropy of X is(66)11−δlogαβ2α−1δ∑i=0∞δα−1ig0−βi+1goβi+δβ−1+1βi+δβ−1+1;δ>0andδ≠1.Proof.
The Rényi entropy forX is defined by
Using Equation (10), we can write(67)fδx=αβg0β2α−1δxδβ−11+xg0βδα−1.
Then, integrationfδx gives(68)∫0gofδxdx=αβ2α−1δ∑i=0∞δα−1igo−βi+1goβi+δβ−1+1βi+δβ−1+1.
Hence, the Rényi entropy reduces to(69)IδX=11−δlogαβ2α−1δ∑i=0∞δα−1ig0−βi+1goβi+δβ−1+1βi+δβ−1+1.
The expression developed in (68) is quite helpful in the further computation of entropy measures of Havrda and Charvat, and Mathai and Haubold. The final expressions of Havrda and Charvat, and Mathai and Haubold entropy measures are presented in Table 2.
Table3 presents the results of Rényi, Havrda and Charvat, and Mathai and Haubold entropy measures for some choices of model parameters for (go  = 3), Set-I (α=1.1,β=2.1), Set -II (α=2.1,β=2.5), Set –III (α=0.1,β=0.6), and Set-IV (α=0.01,β=1.1).
A wide range of positive and negative values of entropy measures makes the EPF distribution more flexible and versatile.Table 2
The final expressions for Havrda and Charvat, and Mathai and Haubold entropy measures.
Entropy/SupportMeasureExpressionHavrda and Charvatω>0,ω≠11/1−ω∫0gofωxdx−111−ωαβ2α−1ω∑i=0∞ωα−1ig0−βi+1×goβi+ωβ−1+1βi+ωβ−1+1−1Mathai and Hauboldϑ>0,ϑ≠11/ϑ−1∫0gof2−ϑxdx−11ϑ−1αβ2α−12−ϑ∑i=0∞2−ϑα−1ig0−βi+1×goβi+2−ϑβ−1+1βi+2−ϑβ−1+1−1Table 3
Numerical analysis for Rényi, Havrda and Charvat, and Mathai and Haubold entropy measures.
EntropyInt.Set -ISet -IISet -IIISet-IVRényiδ=1.148.507345.556842.999936.4934δ=1.513.229212.424611.72729.9527δ=1.710.709410.05809.49348.0570δ=1.99.30948.74328.25257.0038Havrda and Charvatω=1.120.821420.913020.745021.0333ω=1.54.66574.74724.49614.8391ω=1.73.46373.53893.23073.6180ω=1.92.77852.84732.45102.9149Mathai and Hauboldℶ=1.1−19.0736−18.9800−19.1095−18.8459ℶ=1.5−2.7702−2.6883−2.7356−2.5421ℶ=1.7−1.3975−1.3360−1.3444−1.2080ℶ=1.9−0.4388−0.4138−0.4027−0.3487


## 5.4. Distribution of Order Statistics

LetX1,…,Xn be a random sample of size n and their corresponding order statistics (OS) X1<...<Xn from the EPF distribution. The PDF of the i-th OS is fi:nx=1/Bi,n−i+1!Fxi−11−Fxn−ifx,i = 1, 2, 3, …, n.Thei-th OS density is obtained by incorporating Equations (6) and (8) in the last equation.(70)fi:nx=1Bi,n−i+1!1+x/g0βα−12α−1i−12α−1+x/g0βα2α−1n−iαβg0β2α−1xβ−11+x/g0βα−1.The minimum and maximum OS densities are obtained, respectively, by substitutingi=1,n in (70).Thei-th OS CDF is defined by(71)Gx=∑r=innr1−Fxn−rFrx.Thei-th OS CDF of the EPF distribution reduces to(72)Gx=∑r=1nnr1+x/g0βα−12α−1r1+x/g0βα−12α−1n−r.The mediani-th OS PDF is(73)fm+1:nx=2m+1!m!2fxFxm1−Fxm.TheXm+1 median OS PDF of the EPF distribution has the form:(74)fm+1:nx=2m+1!m!2αβg0β2α−1xβ−11+xg0βα−11+x/g0βα−12α−1m2α−1+x/g0βα2α−1m.Thei-th and j-th OS joint distribution is defined by(75)fxi,xj=CFxii−1Fxj−Fxij−i−11−Fxjn−jfxifxj.For thei-th and j-th OS joint distribution of the EPF model is as follows:(76)fxi,xj=Cαβ2xixjβ−1g02β2α−121+xig0β1+xjg0βα−11+x/g0βα−12α−1i−11−1+x/g0βα−12α−1n−j1+xj/g0βα−1+xi/g0βα2α−1j−i−1.


## 5.5. Bivariate and Multivariate Extensions

In this section, we develop the bivariate and multivariate extensions for the EPF distribution by following the Morgenstern family and the Clayton familyThe CDF of the bivariate EPF distribution followed by the Clayton family for the random vectorX,Y is(77)FV1,V2=1+η1−Z1v11−Z2v2Z1v1Z2v2,whereη≤1, Z1v1=1+x1/g0β1α1−1/2α1−1 and Z2v2=1+x2/g0β2α2−1/2α2−1.The CDF of the bivariate EPF distribution followed by the Morgenstern family for the random vectorV1,V2 is defined as(78)Cx,y=x−τ1+τ2+y−τ1+τ2−1−1/τ1+τ2;τ1+τ2≥0.Let v1∼ EPF (α1,β1), and v2∼ EPF (α2,β2). Then, we set x=Z1v1=1+x1/g0β1α1−1/2α1−1, and y=Z2v2=1+x2/g0β2α2−1/2α2−1.The CDF of the bivariate EPF distribution followed by the Clayton family for the random vectorV1,V2 is(79)Gv1,v2=1+x1/g0β1α1−12α1−1τ1+τ2+1+x2/g0β2α2−12α2−1τ1+τ2−1−1/τ1+τ2.A simplen-dimensional extension of the last version for EPF distribution has the form:(80)Hx1,x2,x3,…,xn=∑j=1n11+x1/g0βjαj−12αj−1τ1+τ2+1−n−1/τ1+τ2.


## 6. Statistical Inference

In this section, we discuss six estimation techniques for the EPF parameters as follows: maximum likelihood estimators (MLEs), maximum product of spacing estimators (MPSEs), percentile estimators (PCEs), Cramér von–Mise distance estimators (CVMEs), Anderson–Darling estimators (ADEs), and right-tail Anderson–Darling estimators (RTADEs).


### 6.1. Maximum Likelihood Estimators

Letx1,…,xn be a random sample of size n from the EPF model; then, the likelihood function of Lϕ=∏j=1nfxj is given by(81)Lϕ=αβg02α−1n∏j=1nxjg0β−1∏j=1n1+xjg0βα−1.The logLϕ  =  lϕ takes the form:(82)lϕ=nlogα+nlogβ−nlogg0−nlog2α−1+β−1∑j=1nlogxjg0+α−1∑j=1nlog1+xjg0β.Letyj=xj/g0. The partial derivatives for the parameters α and β are(83)∂lϕ∂α=nα−n2αlog22α−1+∑j=1nlog1+yjβ, and (84)∂lϕ∂β=nβ+∑j=1nlogyj+α−1∑j=1nyβlogyj1+yjβ.The maximum likelihood estimates (ϕ^=α,^β^) of the EPF parameters can be obtained by maximizing (82) or by solving the above nonlinear equations simultaneously. These nonlinear equations although do not provide an analytical solution for the MLEs and the optimum values of α, and β. Consequently, the Newton–Raphson type algorithm is an appropriate choice to obtain the MLEs.


### 6.2. Maximum Product of Spacing Estimators

The MPSEs are alternatives to the MLEs, and they are introduced by Cheng and Amin [49, 50]. Let x1,…,xn be a uniform spacing of a random sample taken from the EPF distribution is defined by Dj=Fxj:n−Fxj−1:n,j=1,2,3,…,n, where Dj denotes the uniform spacing, xj is j-th order statistics, and x0:n=0,Fxn+1:n=1, and ∑j=1n+1Dj=1, and the MPSEs of the EPF parameters are obtained by maximizing(85)Pα,β=1n+1∑j=1n+1logDj.These estimators can also be obtained by solving(86)1n+1∑j=1n+11Dj∇txj:n−∇txj−1:n=0,where(87)∇1xj:n=∂∂αFxj:n,∇2xj:n=∂∂βFxj:n.


### 6.3. Percentile Estimators

The percentile method was introduced by Kao [51]. This method allows estimating the unknown parameters if the distribution function has a closed-form expression. Suppose uj=j/n+1 be an unbiased estimator of Fxj:n. The PCEs of the EPF parameters are obtained by minimizing(88)Pα,β=∑j=1nxj:n−g02α−1uj+11/α−11/β2,with respect to α and β, respectively.


### 6.4. Cramér von–Mise Estimators

Cramér [52] and Von Mises [53] introduced a relatively less-biased minimum distance estimator called the CVMEs. It can be obtained by making a difference between the estimates of the CDF and empirical CDF. The CVMEs of the EPF parameters are obtained by minimizing(89)Cα,β=112n+∑j=1nFxj:n−2j−12n,with respect to α and β. Furthermore, the CVMEs follow by solving the nonlinear equations as(90)Cα,β=112n+∑j=1nFxj:n−2j−12n∇txj:n=0,where ∇txj:n=0 for t = 1, 2 is defined by (87).


### 6.5. Anderson–Darling and Right-Tail Anderson–Darling Estimators

Another type of minimum distance estimators is the ADEs. The ADEs of the EPF parameters are obtained by minimizing(91)Aα,β=−n−1n+∑j=1n2j−1logFxj:n+logSxj:n,with respect to α and β, respectively. The ADEs are also obtained by solving the following nonlinear equation as(92)∑j=1n2j−1∇txj:nFxj:n+∇txn+1−j:nSxn+1−j:n,where ∇txj:n=0 (for t = 1, 2) is defined by (87).The RTADEs of the EPF parameters can be determined by minimizing(93)Aα,β=n2−2+∑j=1nFxj:n−1n∑j=1n2j−1logSxj:n,with respect to α and β.


## 6.1. Maximum Likelihood Estimators

Letx1,…,xn be a random sample of size n from the EPF model; then, the likelihood function of Lϕ=∏j=1nfxj is given by(81)Lϕ=αβg02α−1n∏j=1nxjg0β−1∏j=1n1+xjg0βα−1.The logLϕ  =  lϕ takes the form:(82)lϕ=nlogα+nlogβ−nlogg0−nlog2α−1+β−1∑j=1nlogxjg0+α−1∑j=1nlog1+xjg0β.Letyj=xj/g0. The partial derivatives for the parameters α and β are(83)∂lϕ∂α=nα−n2αlog22α−1+∑j=1nlog1+yjβ, and (84)∂lϕ∂β=nβ+∑j=1nlogyj+α−1∑j=1nyβlogyj1+yjβ.The maximum likelihood estimates (ϕ^=α,^β^) of the EPF parameters can be obtained by maximizing (82) or by solving the above nonlinear equations simultaneously. These nonlinear equations although do not provide an analytical solution for the MLEs and the optimum values of α, and β. Consequently, the Newton–Raphson type algorithm is an appropriate choice to obtain the MLEs.


## 6.2. Maximum Product of Spacing Estimators

The MPSEs are alternatives to the MLEs, and they are introduced by Cheng and Amin [49, 50]. Let x1,…,xn be a uniform spacing of a random sample taken from the EPF distribution is defined by Dj=Fxj:n−Fxj−1:n,j=1,2,3,…,n, where Dj denotes the uniform spacing, xj is j-th order statistics, and x0:n=0,Fxn+1:n=1, and ∑j=1n+1Dj=1, and the MPSEs of the EPF parameters are obtained by maximizing(85)Pα,β=1n+1∑j=1n+1logDj.These estimators can also be obtained by solving(86)1n+1∑j=1n+11Dj∇txj:n−∇txj−1:n=0,where(87)∇1xj:n=∂∂αFxj:n,∇2xj:n=∂∂βFxj:n.


## 6.3. Percentile Estimators

The percentile method was introduced by Kao [51]. This method allows estimating the unknown parameters if the distribution function has a closed-form expression. Suppose uj=j/n+1 be an unbiased estimator of Fxj:n. The PCEs of the EPF parameters are obtained by minimizing(88)Pα,β=∑j=1nxj:n−g02α−1uj+11/α−11/β2,with respect to α and β, respectively.


## 6.4. Cramér von–Mise Estimators

Cramér [52] and Von Mises [53] introduced a relatively less-biased minimum distance estimator called the CVMEs. It can be obtained by making a difference between the estimates of the CDF and empirical CDF. The CVMEs of the EPF parameters are obtained by minimizing(89)Cα,β=112n+∑j=1nFxj:n−2j−12n,with respect to α and β. Furthermore, the CVMEs follow by solving the nonlinear equations as(90)Cα,β=112n+∑j=1nFxj:n−2j−12n∇txj:n=0,where ∇txj:n=0 for t = 1, 2 is defined by (87).


## 6.5. Anderson–Darling and Right-Tail Anderson–Darling Estimators

Another type of minimum distance estimators is the ADEs. The ADEs of the EPF parameters are obtained by minimizing(91)Aα,β=−n−1n+∑j=1n2j−1logFxj:n+logSxj:n,with respect to α and β, respectively. The ADEs are also obtained by solving the following nonlinear equation as(92)∑j=1n2j−1∇txj:nFxj:n+∇txn+1−j:nSxn+1−j:n,where ∇txj:n=0 (for t = 1, 2) is defined by (87).The RTADEs of the EPF parameters can be determined by minimizing(93)Aα,β=n2−2+∑j=1nFxj:n−1n∑j=1n2j−1logSxj:n,with respect to α and β.


## 7. Simulation Experiment

In this section, we perform a simulation study to assess the behavior of different estimators in estimating the EPF parameters. We generateN = 1,000 replicates using (61) for several sample sizes n = 25, 50, and 100 with different combinations of the parameters. We calculate the average values of the estimates (AEs):(94)AEφ^=1N∑i=1Nφ^,and the mean square errors (MSEs):(95)MSEφ^=1N∑i=1Nφ^i−φ2,where φ=α,β.The selection of the best estimation method will be made having a minimum value of MSEs. TheR software (DEoptim package) is adopted to obtain the simulation results. The results of AEs and MSEs (in parenthesis) for the MLEs, MPSEs, PCEs, CVMEs, ADEs, and RTADEs are presented in Tables 4–7. It is noted that the AEs tend to their true parameter values, and the MSEs decrease with the increase in the sample size. This evidence is enough to favor that the estimators are unbiased asymptotically. All estimation methods perform efficiently for different combinations.Table 4
The AEs and MSEs forα=1.5andβ=2.0T.
nPar.Est.MLEsMPSEsPCEsCVMEsADEsRTADEs25α^AE1.5430 (0.0808)1.5267 (0.0701)1.5072 (0.0624)2.1794 (0.0910)1.5603 (0.0910)1.5871 (0.1243)MSEβ^AE2.0408 (0.3065)2.0101 (0.2620)2.7331 (2.0871)2.2208 (0.2764)2.0208 (0.2764)2.0156 (0.2693)MSE50α^AE1.5380 (0.0658)1.5225 (0.0631)1.5051 (0.0615)1.7521 (0.0739)1.5474 (0.0704)1.6042 (0.0988)MSEβ^AE2.0080 (0.2592)2.0020 (0.2522)2.8602 (2.2141)2.1503 (0.2278)2.0027 (0.2530)2.0026 (0.2529)MSE100α^AE1.5048 (0.0609)1.5037 (0.0610)1.5001 (0.0575)1.5407 (0.0086)1.5053 (0.0610)1.5179 (0.0597)MSEβ^AE2.0009 (0.2509)2.0005 (0.2505)2.6984 (1.6397)2.0602 (0.0541)2.0006 (0.2506)2.0011 (0.2511)MSETable 5
The AEs and MSEs forα=2.5andβ=1.5T.
nPar.Est.MLEsMPSEsPCEsCVMEsADEsRTADEs25α^AE2.5273 (0.6168)2.5153 (0.5912)2.5001 (0.5627)2.7881 (1.6927)2.5425 (0.6557)2.5900 (0.8361)MSEβ^AE1.5308 (0.0105)1.5100 (0.0024)2.1981 (0.8746)1.8852 (1.3466)1.5182 (0.0051)1.5156 (0.0031)MSE50α^AE2.5270 (0.6001)2.5394 (0.5378)2.5001 (0.5627)2.6770 (1.2651)2.5448 (0.6367)2.5675 (0.7333)MSEβ^AE1.5188 (0.0055)1.5047 (0.0008)2.2336 (0.7808)1.7453 (0.8573)1.5088 (0.0024)1.5088 (0.0025)MSE100α^AE2.5249 (0.5734)2.5197 (0.4719)2.5000 (0.5626)2.5033 (0.5659)2.5241 (0.6150)2.5281 (0.6267)MSEβ^AE1.5032 (0.0001)1.5006 (0.0000)2.2794 (0.7528)1.5003 (0.0000)1.5018 (0.0001)1.5018 (0.0001)MSETable 6
The AEs and MSEs forα=0.8andβ=0.5T.
nPar.Est.MLEsMPSEsPCEsCVMEsADEsRTADEs25α^AE0.8861 (0.5333)0.8802 (0.5386)0.8022 (0.3627)0.8902 (0.6356)0.8968 (0.5663)0.9439 (0.6981)MSEβ^AE0.5156 (0.0019)0.5026 (0.0003)0.5355 (0.0107)0.5325 (0.0015)0.5105 (0.0018)0.5125 (0.0021)MSE50α^AE0.8229 (0.4093)0.8267 (0.4108)0.8026 (0.3531)0.8377 (0.5548)0.8238 (0.4121)0.8348 (0.4369)MSEβ^AE0.5083 (0.0007)0.5036 (0.0002)0.5074 (0.0016)0.5234 (0.0007)0.5049 (0.0005)0.5039 (0.0004)MSE100α^AE0.8076 (0.3702)0.8061 (0.3677)0.8023 (0.3227)0.8089 (0.4527)0.8089 (0.3721)0.8243 (0.3958)MSEβ^AE0.5021 (0.0000)0.5008 (0.0000)0.5006 (0.0000)0.5110 (0.0001)0.5008 (0.0000)0.5005 (0.0000)MSETable 7
The AEs and MSEs forα=2.5andβ=3.5T.
nPar.Est.MLEsMPSEsPCEsCVMEsADEsRTADEs25α^AE2.5676 (0.2944)2.5444 (0.2493)3.1695 (2.0854)2.7040 (0.7616)2.5868 (0.3311)2.6072 (0.3467)MSEβ^AE3.5951 (0.3132)3.5293 (0.1982)4.6798 (2.7572)3.5902 (0.3670)3.5542 (0.2382)3.5717 (0.2741)MSE50α^AE2.5186 (0.1813)2.5082 (0.1685)2.9012 (1.1522)2.5475 (0.2712)2.5375 (0.2088)2.5907 (0.3245)MSEβ^AE3.5207 (0.1824)3.5049 (0.1645)3.7910 (1.9981)3.5154 (0.1948)3.5075 (0.1674)3.5087 (0.1688)MSE100α^AE2.5035 (0.1636)2.5025 (0.1623)2.7081 (0.8632)2.5011 (0.1609)2.5018 (0.1616)2.5035 (0.1636)MSEβ^AE3.5035 (0.1633)3.5009 (0.1608)3.5679 (1.6502)3.5003 (0.1602)3.5020 (0.1618)3.5020 (0.1618)MSE


## 8. Application in Automobile Engineering

In this section, we analyze automobile engineering data. The data represent the time to failure (103 h) of turbocharger of one type of engine discussed by Xu et al. [54]. The observations are as follows: 1.6, 2.0, 2.6, 3.0, 3.5, 3.9, 4.5, 4.6, 4.8, 5.0, 5.1, 5.3, 5.4, 5.6, 5.8, 6.0, 6.0, 6.1, 6.3, 6.5, 6.5, 6.7, 7.0, 7.1, 7.3, 7.3, 7.3, 7.7, 7.7, 7.8, 7.9, 8.0, 8.1, 8.3, 8.4, 8.4, 8.5, 8.7, 8.8, 9.0. This dataset is analyzed by Afify et al. [55] and Nassar et al. [56].The EPF distribution is compared with some well-known competitors, namely, the Weibull power function (W-PF) and the zero-truncated Poisson power function (ZTP-PF). Their CDFs are presented in Table8. The criterion -log-likelihood (-LL), Akaike information criterion (AIC), along with the goodness-of-fit statistics such as Kolmogorov –Smirnov (KS) with its p-value, are adopted. Some descriptive statistics are presented in Table 9. Table 10 presents the estimates and standard errors (SEs) alongside the goodness-of-fit statistics as well. Based on the results in Table 10, we conclude that the EPF distribution provide better fit among all well-established competitors.Table 8
List of some competitive models of CDFs.
ModelCDFs of modelAuthor (s)W-PFGIx=1−e−αxβ/g0β−xβγ, α,β,γ>0, 0<x<g0Tahir et al. [10]ZTP-PFGIIx=1−e−αx/g0β1−e−α, α,β>0, 0<x<g0Okorie et al. [58]Table 9
Descriptive statistics for turbocharger data.
NatureMin.Q1MedianMeanQ3Max.Sk.Kur.Turbocharger1.6005.0756.5006.2537.8259.000−0.6382.641Table 10
Parameter estimates, SEs, and goodness-of-fit statistics for turbocharger data.
Parameter estimates (SEs)StatisticsModelα^β^γ^-LLAICKSp value (KS)EPF0.2516 (0.6067)2.6418 (0.7979)−76.9082157.81660.04961.0000W-PF1.2327 (0.7999)3.1875 (1.7180)0.7639 (0.2057)77.6748161.41640.05150.9999ZTP-PF2.7724 (0.5035)3.8728 (0.5176)−82.4755168.95100.12530.5555Furthermore, the empirical fitted density (i), estimated CDF (ii), probability-probability (PP) (iii), Kaplan–Meier survival (iv), along with the TTT plot (v), and box plot (vi) are illustrated in Figure4, respectively. All the estimates and numerical results are calculated using the statistical software R, package adequacy model developed by Rafael et al. [57].Figure 4
Fitted density (i), estimated CDF (ii), PP plot (iii), Kaplan–Meier survival (iv), TTT plot (v), and box plot (vi) for turbocharger data.


## 9. Conclusions

In this paper, we develop a new class that generates optimal univariate continuous models called the new exponentiated-G class. A special member of the proposed family called the exponentiated power function (EPF) distribution is studied in detail. Numerous statistical and reliability characteristics are discussed. Furthermore, the EPF distribution has flexible shapes for its density and hazard functions. For the estimation of EPF parameters, we followed the six accredited techniques named, MLEs, MPSEs, PCEs, CVMEs, ADEs, and RTADEs. A simulation experiment is performed to compare the performance of different estimation techniques. Our results show that the estimation techniques perform very well. The applicability of the EPF distribution is addressed using real-life data form the engineering field. The results show that the EPF distribution provides better fit as compared to other well-known competitors.For some possible future studies, the EPF distribution can be adopted to analyze entropy measures following the works of Siddiqui et al. [59] and Rashid et al. [60].



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