# Odd Inverse Power Generalized Weibull Generated Family of Distributions: Properties and Applications

Authors: A.S. Al-Moisheer; I. Elbatal; Waleed Almutiry; Mohammed Elgarhy
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A novel family of produced distributions, odd inverse power generalized Weibull generated distributions, is introduced. Various mathematics structural properties for the odd inverse power generalized Weibull generated family are computed. Numerical analysis for mean, variance, skewness, and kurtosis is performed. The new family contains many new models, and the densities of the new models can be right skewed and symmetric with “unimodal” and “bimodal” shapes. Also, its hazard rate function can be “constant,” “decreasing,” “increasing,” “increasing-constant,” “upside-down-constant,” and “decreasing-constant.” Different types of entropies are calculated. Some numerical values of various entropies for some selected values of parameters for the odd inverse power generalized Weibull exponential model are computed. The maximum likelihood estimation, least square estimation, and weighted least square estimation approaches are used to estimate the OIPGW-G parameters. Many bivariate and multivariate type models have been also derived. Two real-world data sets are used to demonstrate the new family’s use and versatility.


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## Body

## 1. Introduction

There are many applications for inverse (I) distributions, including econometrics, biology, life testing, engineering sciences as well as medical investigation and survey sampling concerns. It is also used in financial literature, environmental studies, survival and dependability theory, and other fields. By applying the inverse transformation to well-known random variables that display distinct properties of density and hazard rate shape, several writers have explored (lifetime) phenomena that a noninverted distribution cannot examine, for example, the I exponential model by [1], I Rayleigh distribution by [2], I Lindley distribution by [3], I power Lindley distribution by [4], I Kumaraswamy distribution by [5], Nadarajah–Haghighi distribution by [6], and I Topp Leone by [7], among others.The author in [8] introduced a new three-parameter distribution called I power generalized Weibull distribution. The distribution function (CDF) of I power generalized Weibull (IPGW) distribution is given by(1)Gλ,α,θz=exp1−1+θz−αλ,z>0,where θ>0 is a scale parameter and α,λ>0 are shape parameters. The associated density function (PDF) is as follows:(2)gλ,α,θz=θαλz−α−11+θz−αλ−1exp1−1+θz−αλ.Recently, statisticians have been interested in proposing new families of univariate distributions that are derived from existing ones. Adding one or more form parameters results in these new generators, which improve accuracy and flexibility in modeling for a variety of diverse real-life applications. The most recent families of distributions to appear in the literature are as follows: a method for introducing a parameter into a family of distributions by [9], beta-G by [10], odd Nadarajah–Haghighi-G by [11], the odd Lindley-G by [12], and the odd Fréchet-G by [13], odd generalized exponential-G by [14], exponentiated power generalized Weibull power series family of distributions by [15], and odd generalized NH-G by [16], among others.Using [17] T-X concept, we create a new broader and more flexible family of distributions known as the odd I power generalized Weibull-G (OIPGW-G) family.(3)FV¯z=exp1−1+θOξ¯αzλ|V¯=λ,α,θ,ξ¯>0andz∈ℝ,where Oξ¯αz=G¯z;ξ¯/Gz;ξ¯α. The corresponding PDF is given by(4)fV¯z=θαλgξ¯zG¯ξ¯zα−11+θOξ¯αzλ−1Gξ¯zα+1exp1+θOξ¯αzλ−1.A random variable (RV)Z with PDF (4) is now indicated as Z∼OIPGW-GV¯. The reliability function (RF) and hazard rate function (HRF) can be derived from each other as F¯V¯z=1−FV¯z and τz;V¯=fV¯z/F¯V¯z. The following is an interpretation of the OIPGW-G family. Let Y be a RV with a continuous G distribution that describes a stochastic system. The probability that a person (or component) will not be working (failure or death) at time z after a lifespan of Y is characterized with Gz;ξ¯/G¯z;ξ¯. The RV Z represents the variability of this chances of failure, and we suppose that it follows the OIPGW-G family of parameters θ,α, and λ, and the CDF of Z it is possible to write(5)PY≤z=PrZ≤Gξ¯zG¯ξ¯z=FV¯z.As a result of this, OIPGW-G family is used for the following reasons:(i)
Specific models with all sorts of HRFs to be defined(ii)
Better matches than other models with the same baseline distribution that has been generated(iii)
Make kurtosis more flexible than the baseline model(iv)
The pdf can be symmetric or right or left skewed and reversed J shaped. An extremely versatile model, the OIPGW-G family of distributions is able to adapt to a variety of various models when its parameters are altered. The OIPGW-G family of distributions includes the following well-known families as special cases in Table 1.Table 1
Submodels of the OIPGW-G family.
λθαReduced familyAuthors1−−Odd Fréchet-GHaq and Elgarhy (2018)−−1Inverse Nadarajah–Haghighi-GNew1−1Inverse exponential-G familyNew1−2Inverse Rayleigh-GNewThe quantile function (QF)QGu is given by the relation as follows:(6)F−1u=QGu=G−11+1θ1−lnu1/λ−11/α−1.Equation (6) can be used in deriving the Bowley skewness coefficient and the Moors kurtosis coefficient.The following is how the rest of this article is organized. The linear form of the PDF and CDF for the new family is expressed in Section2. Section 3 contains several special models of the OIPGW-G family. Section 4 discusses the new family’s structural characteristics as moments (Mos), incomplete Mos (IMos), mean deviations (MDes), Bonferroni (Bon) and Lorenz (Lor) curves, moment generating function (MoGF), and Probability Weighted Moments (PrWMos). Also, some numerical analyses for the mean M(Z), variance Var(Z), coefficient of skewness CS(Z), coefficient of kurtosis CK(Z), and coefficient of variation CV(Z) are discussed in the same section. Section 5 computes entropies of various sorts and also some numerical values of different entropies for specific selected parameter values for the odd inverse power generalized Weibull exponential model. In Section 6, parameters are estimated using three different techniques as the maximum likelihood (ML), least square (LS), and weighted LS (WLS) techniques. Many bivariate and multivariate type models have also been examined in Section 7. In Section 8, two applications to real-world data sets illustrate the empirical significance of the odd inverse power generalized Weibull exponential model. At the end of the paper, there are conclusions.


## 2. Important Representation

We give a helpful linear form for the OIPGW-G PDF in this section. Ifz1/z2<1 and ∇>0 is a real noninteger, then the next power series (PS) expansions hold.(7)1−z1z2−∇=∑κ=0∞Γ∇+κκ!Γ∇z1z2κ,(8)1−z1z2∇=∑κ=0∞−1κΓ1+∇κ!Γ1+∇−κz1z2κ,and an exponential function is implemented by using a PS, and we get(9)exp−1+θOξ¯αzλ=∑ℏ=0∞1ℏ!−1ℏ1+θOξ¯αzλℏ.Inserting (9) in (4), we have(10)fV¯z=θαλexp1gξ¯z∑ℏ=0∞−1ℏℏ!G¯ξ¯zα−1Gξ¯zα+11+θOξ¯αzλℏ+1−1︸Az.Using (8), we can write(11)Az=∑ℏ=0∞θwΓλ·w!Γλ·−wG¯ξ¯zαwGξ¯zαw|λ·=λℏ+1,inserting (11) into (10). Then, equation (10) can be written as(12)fV¯z=θαλexp1gξ¯z∑ℏ,w=0∞−1ℏθwΓλ·ℏ!w!Γλ·−wG¯ξ¯zα·−1Gξ¯zα·+1|α·=αw+1.Applying (7) to Gξ¯zα·+1, we can write(13)Gξ¯z−α·+1=∑κ=0∞Γα·+κ+1κ!Γα⋅+11−Gξ¯zκ.Substituting (13) into (12) and applying (8), the OIPGW-G PDF is(14)fV¯z=∑m=0∞Δmπm+1z,where(15)Δm=∑ℏ,w,κ=0∞−1ℏθw+1Γλ·Γα·+κ+1Γα·+κℏ!w!κ!m!m+1Γλℏ+1−wΓα·+1Γα·+κ−m,and πm+1z=m+1gzGmz is the exp-G PDF with power parameter m+1. Also, the CDF of the OIPGW-G family is(16)FV¯z=∑m=0∞ΔmΠm+1z,where Πm+1z is the exp-G CDF with power parameter m+1.


## 3. Some Special Models of the OIPGW-G Family

In this part, we described four submodels of the OIPGW-G family of distributions. There is no doubt about it, and the PDF (4) will be the most tractable when the CDF Gz and PDF gz have easy-to-understand analytic expressions. When we start with the baseline distributions: uniform (U), exponential (Ex), Weibull (W), and Rayleigh (R), we get at four submodels of this family. Table 2 shows the CDF and PDF of various baseline models.Table 2
The CDF and PDF of these baseline models.
ModelGξ¯zOξ¯αzCDFUniformz/ββ/z−1αexp1−1+θβ/z−1αλExponential1−exp−βzexpβz−1−αexp1−1+θexpβz−1−αλWeibull1−exp−zβexpzβ−1−αexp1−1+θexpzβ−1−αλRayleigh1−exp−βz2expz2−1−αexp1−1+θexpβz2−1−αλFigures1 and 2 represent the plots of PDFs and the HRFs for the models which are reported in Table 2. From Figure 1, we can note that the PDFs can be right skewed and symmetric with “unimodal” and “bimodal” shapes. The HRFs can be “constant,” “decreasing,” “increasing,” “increasing-constant,” “upside-down-constant,” and “decreasing-constant.”Figure 1
The plots of PDFs for OIPGWU, OIPGWEx, OIPGWW, and OIPGWR distributions.Figure 2
The plots of HRFs for OIPGWU, OIPGWEx, OIPGWW, and OIPGWR distributions.


## 4. Statistical Properties

### 4.1. Ordinary Moments and Incomplete Moments Functions

Therth ordinary Mos of Z, say μr/, is driven from (14) as(17)μr/=EZr=∑m=0∞ΔmZm+1rz,where Zm+1 has a power parameter of m+1 and represents the exp-G random variable. It is possible to obtain the CS and CK measures by using the nth central Mos, say Mnz of Z, where(18)Mnz=EZ−μ1/n=∑r=0nnr−μ1/n−rEZr=∑r=0n∑κ,m=0∞nr−μ1/n−rηκ,mEZm+1r.Also, the cumulantsϰs of Z follow recursively from(19)ϰs=μs/−∑d=0s−1s−1d−1ϰdμs−d/,where ϰ1=μ1′, ϰ2=μ2′−μ1′2,andϰ3=μ3′−3μ2/μ1/+μ1′3.It is possible to determine the MDes, Bon, and Lor curves using the first IMos. It is highly important in economics and dependability as well as in demography as well as in the fields of insurance and medical. Not only in econometrics but also in many other fields as well, this is apparent. Thesth IMos of Z defined by υst for any real s>0 can be investigated from (14) as(20)υst=∫−∞tzsfzdz=∑m=0∞Δm∫−∞tzsπm+1zdz.Equation (20) denotes the sth incomplete moments of Zm+1. As well as providing significant information on population characteristics, the MDes have been used to income fields and property in the field of economics for a long time. If Z has the OIPGW-G family of distribution, then the MDes about the mean μ=EZ and the MDes about the median M are defined by(21)Dμz=EZ−μ1′=2μ1′Fμ1′−2υ1μ1′,DMz=EZ−M=μ1′−2υ1M,respectively, where μ1′=Ez, M=medianz=Q1/2, Fμ1′ is evaluated from (3), and υ1t is the first IMo given by (20) with s=1, where(22)υ1t=∫−∞tzfzdz=∑m=0∞Δm∫−∞tzπm+1zdz.We can determineδμz and δMz by two techniques; the first can be obtained from (14) as υ1t=∑m=0∞ΔmZm+1t where Zm+1t=∫−∞tzπm+1zdz is the first IMo of the exp-G distribution. The second technique is given by υ1t=∑m=0∞Δmγm+1t where(23)γm+1t=m+1∫0Gtum+1QGudu,can be calculated numerically and QGu=G−1u;φ. The Lor and Bon curves, for a given probability p, are given by Lp=1/μ1′υ1q and Bp=1/pμ1′υ1q, respectively, where μ1′=Ez, and q=Qp is the QF of Z at p. Tables 3 and 4 give numerical analysis for M(Z), Var(Z), CS(Z), CK(Z), and CV(Z).Table 3
Numerical values ofMZ, VarZ, CSZ, CKZ, and CVZ at θ=β=α=0.5 for the OIPGWEx model.
λMZVarZCSZCKZCVZ0.20.4723.0036.46660.0133.6690.51.1646.6994.03925.4032.2240.81.8179.5443.18417.051.71.22.61612.3062.61812.6561.3411.53.1613.8032.3711.0191.1762.03.97115.5452.1139.5180.993Table 4
Numerical values ofMZ, VarZ, CSZ, CKZ, and CVZ at θ=β=λ=0.5 for the OIPGWEx model.
αMZVarZCSZCKZCVZ0.31.65318.6054.1926.4042.6090.80.9762.6553.68622.761.671.20.951.2193.17218.8661.1621.50.970.82.83716.3830.9222.01.0160.4682.39913.2920.673Figures3–6 represent 3D plots of MZ, VarZ, CSZ, and CKZ of the OIPWU, OIPWEx, OIPWW, and OIPWR distributions for several values of parameters.Figure 3
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWU distribution for β=θ=2.Figure 4
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWEx distribution for β=θ=2.Figure 5
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWW distribution for β=5 and θ=2.Figure 6
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWR distribution for β=θ=2.From Figures3–6, MZ and VarZ are decreasing for all models, CSZ can be can be negative or positive, indicating left or right skewed, and CKZ can be increasing.The first formula of MoGF may be computed as follows from equation (14):(24)MZt=EetZ=∑m=0∞ΔmMm+1t,where Mm+1t is the MoGF of Zm+1. Consequently, MZt can be easily determined from the exp-G MoGF. A second alternative formula can be computed from (14) as MZt=∑m=0∞Δmδt,m+1 where δt,φ=φ∫01uφ−1etQGudu.


### 4.2. Probability Weighted Moments

Thes,rth PrWMos of the OIPGW-G family is(25)ϒs,r=EZrFzr=∫−∞∞zsFzrfzdz,from (3) and (4), and after some algebra, we get(26)fzFzr=∑m=0∞bmπm+1z,where(27)bm=exp1+rαλ∑ℏ,w,κ=0∞θw+1r+1ℏ−1ℏ+mΓλℏ+1Γα·+κ−1w!κ!m!Γr+2Γλℏ+1−wΓα·−1×Γα·+κΓα·+κ−mm+1.Therefore, thes,rth PrWMos of the OIPGW-G family can be expressed as(28)ϒs,r=∑m=0∞bm∫−∞∞zsπm+1zdz,Thus, thes,rth PrWMos of Z is(29)ϒs,r=∑m=0∞bmEZm+1s.


## 4.1. Ordinary Moments and Incomplete Moments Functions

Therth ordinary Mos of Z, say μr/, is driven from (14) as(17)μr/=EZr=∑m=0∞ΔmZm+1rz,where Zm+1 has a power parameter of m+1 and represents the exp-G random variable. It is possible to obtain the CS and CK measures by using the nth central Mos, say Mnz of Z, where(18)Mnz=EZ−μ1/n=∑r=0nnr−μ1/n−rEZr=∑r=0n∑κ,m=0∞nr−μ1/n−rηκ,mEZm+1r.Also, the cumulantsϰs of Z follow recursively from(19)ϰs=μs/−∑d=0s−1s−1d−1ϰdμs−d/,where ϰ1=μ1′, ϰ2=μ2′−μ1′2,andϰ3=μ3′−3μ2/μ1/+μ1′3.It is possible to determine the MDes, Bon, and Lor curves using the first IMos. It is highly important in economics and dependability as well as in demography as well as in the fields of insurance and medical. Not only in econometrics but also in many other fields as well, this is apparent. Thesth IMos of Z defined by υst for any real s>0 can be investigated from (14) as(20)υst=∫−∞tzsfzdz=∑m=0∞Δm∫−∞tzsπm+1zdz.Equation (20) denotes the sth incomplete moments of Zm+1. As well as providing significant information on population characteristics, the MDes have been used to income fields and property in the field of economics for a long time. If Z has the OIPGW-G family of distribution, then the MDes about the mean μ=EZ and the MDes about the median M are defined by(21)Dμz=EZ−μ1′=2μ1′Fμ1′−2υ1μ1′,DMz=EZ−M=μ1′−2υ1M,respectively, where μ1′=Ez, M=medianz=Q1/2, Fμ1′ is evaluated from (3), and υ1t is the first IMo given by (20) with s=1, where(22)υ1t=∫−∞tzfzdz=∑m=0∞Δm∫−∞tzπm+1zdz.We can determineδμz and δMz by two techniques; the first can be obtained from (14) as υ1t=∑m=0∞ΔmZm+1t where Zm+1t=∫−∞tzπm+1zdz is the first IMo of the exp-G distribution. The second technique is given by υ1t=∑m=0∞Δmγm+1t where(23)γm+1t=m+1∫0Gtum+1QGudu,can be calculated numerically and QGu=G−1u;φ. The Lor and Bon curves, for a given probability p, are given by Lp=1/μ1′υ1q and Bp=1/pμ1′υ1q, respectively, where μ1′=Ez, and q=Qp is the QF of Z at p. Tables 3 and 4 give numerical analysis for M(Z), Var(Z), CS(Z), CK(Z), and CV(Z).Table 3
Numerical values ofMZ, VarZ, CSZ, CKZ, and CVZ at θ=β=α=0.5 for the OIPGWEx model.
λMZVarZCSZCKZCVZ0.20.4723.0036.46660.0133.6690.51.1646.6994.03925.4032.2240.81.8179.5443.18417.051.71.22.61612.3062.61812.6561.3411.53.1613.8032.3711.0191.1762.03.97115.5452.1139.5180.993Table 4
Numerical values ofMZ, VarZ, CSZ, CKZ, and CVZ at θ=β=λ=0.5 for the OIPGWEx model.
αMZVarZCSZCKZCVZ0.31.65318.6054.1926.4042.6090.80.9762.6553.68622.761.671.20.951.2193.17218.8661.1621.50.970.82.83716.3830.9222.01.0160.4682.39913.2920.673Figures3–6 represent 3D plots of MZ, VarZ, CSZ, and CKZ of the OIPWU, OIPWEx, OIPWW, and OIPWR distributions for several values of parameters.Figure 3
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWU distribution for β=θ=2.Figure 4
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWEx distribution for β=θ=2.Figure 5
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWW distribution for β=5 and θ=2.Figure 6
3D plots ofMZ, VarZ, CSZ, and CKZ of the OIPWR distribution for β=θ=2.From Figures3–6, MZ and VarZ are decreasing for all models, CSZ can be can be negative or positive, indicating left or right skewed, and CKZ can be increasing.The first formula of MoGF may be computed as follows from equation (14):(24)MZt=EetZ=∑m=0∞ΔmMm+1t,where Mm+1t is the MoGF of Zm+1. Consequently, MZt can be easily determined from the exp-G MoGF. A second alternative formula can be computed from (14) as MZt=∑m=0∞Δmδt,m+1 where δt,φ=φ∫01uφ−1etQGudu.


## 4.2. Probability Weighted Moments

Thes,rth PrWMos of the OIPGW-G family is(25)ϒs,r=EZrFzr=∫−∞∞zsFzrfzdz,from (3) and (4), and after some algebra, we get(26)fzFzr=∑m=0∞bmπm+1z,where(27)bm=exp1+rαλ∑ℏ,w,κ=0∞θw+1r+1ℏ−1ℏ+mΓλℏ+1Γα·+κ−1w!κ!m!Γr+2Γλℏ+1−wΓα·−1×Γα·+κΓα·+κ−mm+1.Therefore, thes,rth PrWMos of the OIPGW-G family can be expressed as(28)ϒs,r=∑m=0∞bm∫−∞∞zsπm+1zdz,Thus, thes,rth PrWMos of Z is(29)ϒs,r=∑m=0∞bmEZm+1s.


## 5. Entropies

This section is dedicated to obtain the expression for different entropy measures of the OIPGW-G family. The Rényi entropy (RéE), presented by [18], is defined by(30)IRρ=1−ρ−1log∫−∞∞fzρdz,ρ≠1,ρ>0.Using (4), applying the same method of the useful expansion (14) and after some algebra, we get(31)fρz=exp1αθλρ∑m=0∞cmgzρGzm,where(32)cm=∑ℏ,w,κ=0∞θw+1−1ℏ+w+mΓραℏ+1+w+ρΓαρ+w+κ+ρℏ!w!κ!Γραℏ+1+ρΓαρ+w+ραρ+w+κ−ρm.Thus, Ré entropy of OIPGW-G family is given by(33)IRρ=ρ1−ρlogexp1αθλ+11−ρlog∑m=0∞cmI−∞∞z,where I−∞∞z=∫−∞∞gzρGzmdz.The Tsallis entropy (TE) measure (see [19]) is defined by(34)TRρ=1ρ−11−∫−∞∞fzρdz,ρ≠1,ρ>0.The Havrda and Charvat entropy (HaChE) measure (see [20]) is defined by(35)HaChRρ=121−ρ−1∫−∞∞fzρdz1/ρ−1,ρ≠1,ρ>0.The Arimoto entropy (ArE) measure (see [21]) of OIPGW-G is defined by(36)ArRρ=ρ1−ρ∫−∞∞fzρdz1/ρ−1,ρ≠1,ρ>0.Some of the numerical values of RéE, QE, HaChE, and ArE for a few chosen parameter values for the OIPGWEx model are given in Tables5–8.Table 5
Numerical values of RéE, QE, HaChE, and ArE atθ=β=α=ρ=0.5 for the OIPGWE model.
λRéEQEHaChEArE0.21.0271.3421.1571.7920.51.8683.0892.6825.4740.82.2354.1163.8738.3511.22.4994.9775.04111.1711.52.6195.415.68612.72822.7475.8986.4614.595Table 6
Numerical values of RéE, QE, HaChE, and ArE atθ=β=α=0.5 and ρ=1.5 for the OIPGWEx model.
λRéEQEHaChEArE0.2−7.415−79.512−3.469−32.5290.5−0.83−1.028−0.386−0.9560.80.6330.542−0.2370.571.21.4751.043−0.1791.1651.51.8081.19−0.161.35822.1181.306−0.1451.519Table 7
Numerical values of RéE, QE, HaChE, and ArE atθ=β=λ=ρ=0.5 for the OIPGWEx model.
αRéEQEHaChEArE0.32.1453.8463.5397.5440.81.6452.5522.1464.1811.21.4362.1011.7423.2051.51.3041.8381.5262.6832.01.1111.4861.2582.037Table 8
Numerical values of RéE, QE, HaChE, and ArE atθ=β=λ=0.5 and ρ=1.5 for the OIPGWEx model.
αRéEQEHaChEArE0.3−3.933−12.29−1.087−8.1290.80.3070.285−0.2640.2921.20.6120.527−0.2390.5531.50.6280.539−0.2380.56620.5520.482−0.2440.504From Tables3–8, we can note that the values of entropies can be negative or positive.


## 6. Statistical Inference

The parameters of the OIPGW-G are estimated using various methods including ML, LS, and WLS methods of estimation.


### 6.1. Maximum Likelihood Estimation

Letz1,…,zn be a n-sized random sample from the OIPGW-G with parameters α,θ,λ, and ξ¯. Let Ω=α,θ,λ,ξ¯T be p×1 vector of parameters. As an example, the log-likelihood function is defined as follows:(37)Ln=nlogθ+nlogα+nlogλ+∑ℏ=1nloggξ¯zℏ+α−1∑ℏ=1nlogG¯ξ¯zℏ−α+1∑ℏ=1nlogGξ¯zℏ+λ−1∑ℏ=1nlog1+θtℏα+∑ℏ=1n1−1+θtℏαλ,where tℏ=G¯ξ¯zℏ/Gξ¯zℏ. The components of score vector UΩ=∂Ln/∂Ω=∂Ln/∂α,∂Ln/∂θ,∂Ln/∂λ,∂Ln/∂ξ¯Tare given by(38)Uα=∂Ln∂α=nα+∑ℏ=1nlogG¯ξ¯zℏ−∑ℏ=1nlogGξ¯zℏ+λ−1∑ℏ=1nθtℏαlogtℏ1+θtℏα−λ∑ℏ=1nθtℏαlogtℏ1+θtℏαλ−1,Uθ=∂Ln∂θ=nθ+λ−1∑ℏ=1ntℏα1+θtℏα−λ∑ℏ=1ntℏα1+θtℏαλ−1,Uλ=∂Ln∂λ=nλ+∑ℏ=1nlog1+θtℏα−∑ℏ=1n1+θtℏαλlog1+θtℏα,Uξ¯κ=∂Ln∂ξ¯κ=∑ℏ=1ngξ¯′zℏGξ¯zℏ+α−1∑ℏ=1nGξ¯′zℏG¯ξ¯zℏ−α+1∑ℏ=1nGξ¯′zℏGξ¯zℏ+λ−1∑ℏ=1nθαtℏα−11+θtℏα∂tℏ∂ξ¯κ−λ∑ℏ=1nθαtℏα−11+θtℏαλ−1∂tℏ∂ξ¯κ,where gξ¯′zℏ=∂Gξ¯zℏ/∂ξ¯κ,Gξ¯′zℏ=∂Gξ¯zℏ/∂ξ¯κ,andG¯ξ¯′zℏ=∂G¯ξ¯zℏ/∂ξ¯κ and ξ¯κ is the κth element of the vector of parameters ξ¯. The maximum likelihood estimation (MLE) of parameters α,θ,λ, and ξ¯ is obtained by setting Uα=Uθ=Uλ=Uξ¯κ=0 and solving these equations simultaneously to get the MLEΩ^=α^,θ^,λ^,ξ¯^TT of Ω.


### 6.2. Ordinary and Weighted Least Square Estimators

Supposez1,…,zn is a random sample from OIPGW-G with corresponding ordered sample of z1,…,zn. The mean and variance of OIPGW-G are independent of unknown parameter and are as follows: EFZi=i/n+1 and varFZi=in−i+1/n+12n+2, where FZi is cdf of OIPGW-G with be the ith order statistic. Then, LS estimators are obtained by minimizing the SSE:(39)∑i=1nFiz−in+12,with respect to Ω.The WLS estimators ofΩ can be obtained by minimizing the following expression:(40)∑i=1nn+12n+2in−i+1Fiz−in+12,with respect to Ω.


### 6.3. Simulation Outcomes

Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of sizen=50,100, and 200 from the OIPGWEx distribution from the following equation:(41)Qu=−1βln1−1+1θ1−lnu1/λ−11/α−1.Three sets of the parameters are assigned. The MLE, LSE, and WLSE ofα,θ,λ, and β are determined by using MATHCAD (14). Then, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 9–11.Table 9
MLEs, LSEs, WLSEs, and MSEs at Set 1:α=λ=θ=β=0.5 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs500.5190.0420.4260.01140.4910.07610.6290.0710.7060.05960.5860.01950.4650.0620.4790.04180.5090.02870.6220.1280.8950.49030.6320.06511000.4930.0170.4340.00920.5080.00290.6020.0490.6850.05150.5760.01080.470.030.4810.02260.5080.01650.560.0620.7150.13230.6210.03772000.50.00860.4510.00880.4940.00280.5610.0250.6130.04450.5660.00540.4610.0230.4860.01470.5040.00850.5250.0280.6710.12440.5710.0317Table 10
MLEs, LSEs, WLSEs, and MSEs at Set 2:α=θ=β=0.5 and λ=1.2 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs500.5190.0160.5020.00750.5550.01221.3780.1831.0980.01781.1380.04570.4820.0630.5740.02610.4350.0320.5590.0660.5390.04330.3840.03581000.5220.010.5080.005180.5490.01171.2880.1491.1090.01321.1720.02690.4950.0370.5610.01790.4360.02430.510.0260.5270.0260.3910.03442000.5210.00890.5110.002860.5350.0091.2340.1391.1060.01161.2360.02550.5340.0370.5530.01050.4630.01930.5130.0180.5070.01280.3960.0258Table 11
MLEs, LSEs, WLSEs, and MSEs at Set 3:θ=β=0.5 and α=λ=1.2 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs501.3780.1071.3740.08111.3450.05111.2330.2491.3150.20511.1280.04950.3850.0830.3960.05210.3760.04150.4160.0560.3860.02760.3580.02191001.3880.0751.360.06821.3180.02391.1850.2021.2220.19651.1130.02420.3860.0540.4010.05160.3950.03810.4220.0260.3910.02650.3780.01552001.3470.0611.3420.06021.2770.01131.1880.1411.1990.18511.20.01570.3870.0440.4110.0480.4480.02780.4590.0280.4660.02510.4760.0107


## 6.1. Maximum Likelihood Estimation

Letz1,…,zn be a n-sized random sample from the OIPGW-G with parameters α,θ,λ, and ξ¯. Let Ω=α,θ,λ,ξ¯T be p×1 vector of parameters. As an example, the log-likelihood function is defined as follows:(37)Ln=nlogθ+nlogα+nlogλ+∑ℏ=1nloggξ¯zℏ+α−1∑ℏ=1nlogG¯ξ¯zℏ−α+1∑ℏ=1nlogGξ¯zℏ+λ−1∑ℏ=1nlog1+θtℏα+∑ℏ=1n1−1+θtℏαλ,where tℏ=G¯ξ¯zℏ/Gξ¯zℏ. The components of score vector UΩ=∂Ln/∂Ω=∂Ln/∂α,∂Ln/∂θ,∂Ln/∂λ,∂Ln/∂ξ¯Tare given by(38)Uα=∂Ln∂α=nα+∑ℏ=1nlogG¯ξ¯zℏ−∑ℏ=1nlogGξ¯zℏ+λ−1∑ℏ=1nθtℏαlogtℏ1+θtℏα−λ∑ℏ=1nθtℏαlogtℏ1+θtℏαλ−1,Uθ=∂Ln∂θ=nθ+λ−1∑ℏ=1ntℏα1+θtℏα−λ∑ℏ=1ntℏα1+θtℏαλ−1,Uλ=∂Ln∂λ=nλ+∑ℏ=1nlog1+θtℏα−∑ℏ=1n1+θtℏαλlog1+θtℏα,Uξ¯κ=∂Ln∂ξ¯κ=∑ℏ=1ngξ¯′zℏGξ¯zℏ+α−1∑ℏ=1nGξ¯′zℏG¯ξ¯zℏ−α+1∑ℏ=1nGξ¯′zℏGξ¯zℏ+λ−1∑ℏ=1nθαtℏα−11+θtℏα∂tℏ∂ξ¯κ−λ∑ℏ=1nθαtℏα−11+θtℏαλ−1∂tℏ∂ξ¯κ,where gξ¯′zℏ=∂Gξ¯zℏ/∂ξ¯κ,Gξ¯′zℏ=∂Gξ¯zℏ/∂ξ¯κ,andG¯ξ¯′zℏ=∂G¯ξ¯zℏ/∂ξ¯κ and ξ¯κ is the κth element of the vector of parameters ξ¯. The maximum likelihood estimation (MLE) of parameters α,θ,λ, and ξ¯ is obtained by setting Uα=Uθ=Uλ=Uξ¯κ=0 and solving these equations simultaneously to get the MLEΩ^=α^,θ^,λ^,ξ¯^TT of Ω.


## 6.2. Ordinary and Weighted Least Square Estimators

Supposez1,…,zn is a random sample from OIPGW-G with corresponding ordered sample of z1,…,zn. The mean and variance of OIPGW-G are independent of unknown parameter and are as follows: EFZi=i/n+1 and varFZi=in−i+1/n+12n+2, where FZi is cdf of OIPGW-G with be the ith order statistic. Then, LS estimators are obtained by minimizing the SSE:(39)∑i=1nFiz−in+12,with respect to Ω.The WLS estimators ofΩ can be obtained by minimizing the following expression:(40)∑i=1nn+12n+2in−i+1Fiz−in+12,with respect to Ω.


## 6.3. Simulation Outcomes

Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of sizen=50,100, and 200 from the OIPGWEx distribution from the following equation:(41)Qu=−1βln1−1+1θ1−lnu1/λ−11/α−1.Three sets of the parameters are assigned. The MLE, LSE, and WLSE ofα,θ,λ, and β are determined by using MATHCAD (14). Then, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 9–11.Table 9
MLEs, LSEs, WLSEs, and MSEs at Set 1:α=λ=θ=β=0.5 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs500.5190.0420.4260.01140.4910.07610.6290.0710.7060.05960.5860.01950.4650.0620.4790.04180.5090.02870.6220.1280.8950.49030.6320.06511000.4930.0170.4340.00920.5080.00290.6020.0490.6850.05150.5760.01080.470.030.4810.02260.5080.01650.560.0620.7150.13230.6210.03772000.50.00860.4510.00880.4940.00280.5610.0250.6130.04450.5660.00540.4610.0230.4860.01470.5040.00850.5250.0280.6710.12440.5710.0317Table 10
MLEs, LSEs, WLSEs, and MSEs at Set 2:α=θ=β=0.5 and λ=1.2 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs500.5190.0160.5020.00750.5550.01221.3780.1831.0980.01781.1380.04570.4820.0630.5740.02610.4350.0320.5590.0660.5390.04330.3840.03581000.5220.010.5080.005180.5490.01171.2880.1491.1090.01321.1720.02690.4950.0370.5610.01790.4360.02430.510.0260.5270.0260.3910.03442000.5210.00890.5110.002860.5350.0091.2340.1391.1060.01161.2360.02550.5340.0370.5530.01050.4630.01930.5130.0180.5070.01280.3960.0258Table 11
MLEs, LSEs, WLSEs, and MSEs at Set 3:θ=β=0.5 and α=λ=1.2 for the OIPGWEx model.
nMLEsMSEsLSEsMSEsWLSEsMSEs501.3780.1071.3740.08111.3450.05111.2330.2491.3150.20511.1280.04950.3850.0830.3960.05210.3760.04150.4160.0560.3860.02760.3580.02191001.3880.0751.360.06821.3180.02391.1850.2021.2220.19651.1130.02420.3860.0540.4010.05160.3950.03810.4220.0260.3910.02650.3780.01552001.3470.0611.3420.06021.2770.01131.1880.1411.1990.18511.20.01570.3870.0440.4110.0480.4480.02780.4590.0280.4660.02510.4760.0107


## 7. Copula under the OIPGWEx Model

The fact thatC·u,v is a straightforward function of the U marginal CDFs, Fu and Gv, is a property shared by all of the probability distributions in this section. These sorts of joint models are referred to as “Copulas (Co).” Copula techniques are widely used in insurance, econometrics, and finance. They are multivariate distributions with uniform marginals on the interval I0,1=0,1. Using FGM Co, modified FGM Co, Clayton Co, and Rényi’s entropy, we construct several new bivariate type OIPGWEx (BvOIPGWEx) models. Additionally, the MvOIPGWEx type of the multivariate KBX is discussed. These new models, on the other hand, may be the subject of future research efforts.


### 7.1. FGM Copula

For starters, let us look at the combined CDF of two RVsW1,W2 for Morgenstern family [22] which is characterized by the following Cλm,ω|λ∈I−1,1andm,ω∈0,1=1+λm¯ω¯mω. Sett m¯=1−m and ω¯=1−ω. Then, m¯=F¯V1¯w1 and ω¯=F¯V2¯w2. Then, we have Cλm,ω=Cλw1,w2.


### 7.2. Modified FGM Copula

This modified FGM Co (see [23–25]) is a good example of this.(42)CΔm,ω|Δ−1,1=mω1+Δϑm℧ω=mω+Δϑm˜℧ω˜,where ϑm˜/m=ϑm and ℧ω˜/ω=℧ω. ϑm and ℧ω consist of two activities that are inseparable I0,1 where ϑ0=℧0=℧1=ϑ1=0. Let(43)α=inf∂∂mϑm˜:W1m<0,ξ¯=inf∂∂ω℧ω˜:W2ω>0,β=sup∂∂mϑm˜:W1m<0,η=sup∂∂ω℧ω˜:W2ω>0,and minαβ,ξ¯η≥1 where(44)∂∂mϑm˜=ϑm+m∂∂mϑm,W1m=m:m∈I0,1|∂∂mϑm˜exists,W2ω=ω:ω∈I0,1|∂∂ω℧ω˜exists.


#### 7.2.1. Bivariate OIPGWEx-FGM (Type-I) Model

The bivariate OIPGWEx-FGM (Type-I) model may be constructed directly from CΔm,ω=mω+Δϑm˜℧ω˜ where ϑm˜=mF¯V1¯m and ℧ω˜=ωF¯V2¯ω.


#### 7.2.2. Bivariate OIPGWEx-FGM (Type-II) Model

Suppose thatϑm and ℧ω meet all of the criteria mentioned previously, and(45)ϑ∗m|Δ1>0=mΔ11−m1−Δ1,℧∗ω|Δ2>0=ωΔ21−ω1−Δ2.The corresponding bivariate Co (henceforth, bivariate OIPGWEx-FGM (Type-II) Co) can be deduced:(46)CΔ,Δ1,Δ2m,ω|Δ∈I−1,1=mω1+Δϑ∗m℧∗ω.


#### 7.2.3. Bivariate OIPGWEx-FGM (Type-III) Model

For both,ϑm and ℧ω have a look at the form below. You may calculate the CDF of the bivariate CDF OIPGWEx-FGM (Type-III) model using ϑ(m), ℧(ω), ϑ⋅m=mlog1+m¯, and ℧⋅ω=ωlog1+ω¯ which can be computed as(47)CΔm,ω=mω1+Δϑ⋅m℧⋅ω.


#### 7.2.4. Bivariate OIPGWEx-FGM (Type-IV) Model

You may calculate the CDF of the bivariate OIPGWEx-FGM (Type-IV) model using CΔm,ω=ωF−1m+mF−1ω−F−1mF−1ω where F−1m and F−1ω can be computed from (6).


### 7.3. The Bivariate OIPGWEx via Rényi’s Entropy

Following [26], the joint CDF of the bivariate OIPGWEx via RéE can be written as Cm,ω=w2m+w1ω−w1w2. Let m=1−F¯V1¯w1∈I0,1 and ω=1−F¯V2¯w2∈I0,1; the associated bivariate OIPGWEx will be Cw1,w2=Cm,ω.


### 7.4. The Bivariate OIPGWEx Extension via Clayton Copula

Bivariate extension through Clayton Copula is a weighted variant of the Clayton Co, which has the formCm,ω|η≥0=m−η+ω−η−1−1/η where m=1−F¯V1¯w1∈I0,1 and ω=1−F¯V2¯w2∈I0,1. Similarly, the multivariate OIPGWE extension can be obtained from Cw1,w2,…,wD=∑i=1Dmi−η+1−D−1/η where mi=1−F¯Vi¯wi∈I0,1.


## 7.1. FGM Copula

For starters, let us look at the combined CDF of two RVsW1,W2 for Morgenstern family [22] which is characterized by the following Cλm,ω|λ∈I−1,1andm,ω∈0,1=1+λm¯ω¯mω. Sett m¯=1−m and ω¯=1−ω. Then, m¯=F¯V1¯w1 and ω¯=F¯V2¯w2. Then, we have Cλm,ω=Cλw1,w2.


## 7.2. Modified FGM Copula

This modified FGM Co (see [23–25]) is a good example of this.(42)CΔm,ω|Δ−1,1=mω1+Δϑm℧ω=mω+Δϑm˜℧ω˜,where ϑm˜/m=ϑm and ℧ω˜/ω=℧ω. ϑm and ℧ω consist of two activities that are inseparable I0,1 where ϑ0=℧0=℧1=ϑ1=0. Let(43)α=inf∂∂mϑm˜:W1m<0,ξ¯=inf∂∂ω℧ω˜:W2ω>0,β=sup∂∂mϑm˜:W1m<0,η=sup∂∂ω℧ω˜:W2ω>0,and minαβ,ξ¯η≥1 where(44)∂∂mϑm˜=ϑm+m∂∂mϑm,W1m=m:m∈I0,1|∂∂mϑm˜exists,W2ω=ω:ω∈I0,1|∂∂ω℧ω˜exists.


### 7.2.1. Bivariate OIPGWEx-FGM (Type-I) Model

The bivariate OIPGWEx-FGM (Type-I) model may be constructed directly from CΔm,ω=mω+Δϑm˜℧ω˜ where ϑm˜=mF¯V1¯m and ℧ω˜=ωF¯V2¯ω.


### 7.2.2. Bivariate OIPGWEx-FGM (Type-II) Model

Suppose thatϑm and ℧ω meet all of the criteria mentioned previously, and(45)ϑ∗m|Δ1>0=mΔ11−m1−Δ1,℧∗ω|Δ2>0=ωΔ21−ω1−Δ2.The corresponding bivariate Co (henceforth, bivariate OIPGWEx-FGM (Type-II) Co) can be deduced:(46)CΔ,Δ1,Δ2m,ω|Δ∈I−1,1=mω1+Δϑ∗m℧∗ω.


### 7.2.3. Bivariate OIPGWEx-FGM (Type-III) Model

For both,ϑm and ℧ω have a look at the form below. You may calculate the CDF of the bivariate CDF OIPGWEx-FGM (Type-III) model using ϑ(m), ℧(ω), ϑ⋅m=mlog1+m¯, and ℧⋅ω=ωlog1+ω¯ which can be computed as(47)CΔm,ω=mω1+Δϑ⋅m℧⋅ω.


### 7.2.4. Bivariate OIPGWEx-FGM (Type-IV) Model

You may calculate the CDF of the bivariate OIPGWEx-FGM (Type-IV) model using CΔm,ω=ωF−1m+mF−1ω−F−1mF−1ω where F−1m and F−1ω can be computed from (6).


## 7.2.1. Bivariate OIPGWEx-FGM (Type-I) Model

The bivariate OIPGWEx-FGM (Type-I) model may be constructed directly from CΔm,ω=mω+Δϑm˜℧ω˜ where ϑm˜=mF¯V1¯m and ℧ω˜=ωF¯V2¯ω.


## 7.2.2. Bivariate OIPGWEx-FGM (Type-II) Model

Suppose thatϑm and ℧ω meet all of the criteria mentioned previously, and(45)ϑ∗m|Δ1>0=mΔ11−m1−Δ1,℧∗ω|Δ2>0=ωΔ21−ω1−Δ2.The corresponding bivariate Co (henceforth, bivariate OIPGWEx-FGM (Type-II) Co) can be deduced:(46)CΔ,Δ1,Δ2m,ω|Δ∈I−1,1=mω1+Δϑ∗m℧∗ω.


## 7.2.3. Bivariate OIPGWEx-FGM (Type-III) Model

For both,ϑm and ℧ω have a look at the form below. You may calculate the CDF of the bivariate CDF OIPGWEx-FGM (Type-III) model using ϑ(m), ℧(ω), ϑ⋅m=mlog1+m¯, and ℧⋅ω=ωlog1+ω¯ which can be computed as(47)CΔm,ω=mω1+Δϑ⋅m℧⋅ω.


## 7.2.4. Bivariate OIPGWEx-FGM (Type-IV) Model

You may calculate the CDF of the bivariate OIPGWEx-FGM (Type-IV) model using CΔm,ω=ωF−1m+mF−1ω−F−1mF−1ω where F−1m and F−1ω can be computed from (6).


## 7.3. The Bivariate OIPGWEx via Rényi’s Entropy

Following [26], the joint CDF of the bivariate OIPGWEx via RéE can be written as Cm,ω=w2m+w1ω−w1w2. Let m=1−F¯V1¯w1∈I0,1 and ω=1−F¯V2¯w2∈I0,1; the associated bivariate OIPGWEx will be Cw1,w2=Cm,ω.


## 7.4. The Bivariate OIPGWEx Extension via Clayton Copula

Bivariate extension through Clayton Copula is a weighted variant of the Clayton Co, which has the formCm,ω|η≥0=m−η+ω−η−1−1/η where m=1−F¯V1¯w1∈I0,1 and ω=1−F¯V2¯w2∈I0,1. Similarly, the multivariate OIPGWE extension can be obtained from Cw1,w2,…,wD=∑i=1Dmi−η+1−D−1/η where mi=1−F¯Vi¯wi∈I0,1.


## 8. Real Data Applications

Comparison of the OIPGWEx distribution with various competing models will be made particularly exponential (Ex), odd Lindley Ex (OLEx), Marshall–Olkin (MO) Ex (MOEx), moment exponential (MEx), the logarithmic Burr–Hatke Ex (LBHEx), generalized MO Ex (GMOEx), beta Ex (BEx), MO Kumaraswamy Ex (MOKwEx), Kumaraswamy exponential (KwEx), the Burr X Ex (BrXEx), and Kw MO Ex (KwMOEx). Some details related to these competitive models are available in [27]. For comparing models, we consider the Cramér-Von Mises (C·) and the Akaike information criterion (AIC) (C1), Bayesian IC (C2), Anderson–Darling (HTML translation failed), Consistent AIC (C3), the Kolmogorov–Smirnov (K. S.), Hannan–Quinn IC (C4), and p value (P. V.) statistics.


### 8.1. Modeling Failure (Relief) Times

The failure time data are the first data set. Recent analyses of these data were conducted by [27]. Table 12 lists the MLEs. Table 13 lists C1, C2, C3, C4, A·, C·, K. S., and P. V. The total time test (TTT) plot for the relief time data, as well as the related box plot, is shown in Figure 7. Based on Figure 7, the HRF of the relief times is “increasing HRF,” and these data have only one EV observation. Figure 8 gives the E-PDF, E-CDF, E-HRF, and P-P plot for relief time data. Figure 9 gives Kaplan–Meier (KM) survival plot for relief time data.Table 12
MLEs values for the first data.
ModelsMLEsExβ0.526OLExβ0.6044MExβ0.950LBHExβ0.5263MOExα,β54.474, 2.316BrXExa,β1.1635, 0.3207GMOExλ,α,β0.519, 89.462, 3.169KwExα,θ,β83.756, 0.568, 3.330BExα,θ,β81.633, 0.542, 3.514MOKwExλ,α,θ,β0.13, 33.24, 0.57, 1.67KwMOExλ,α,θ,β8.87, 34.83, 0.29, 4.89OIPGWExλ,α,θ,β1.971, 2.36, 0.51, 0.47Table 13
C1, C2, C3, C4, A·, C·, K. S., and P. V. for the first data.
ModelsC1, C2, C3, C4A·C·K. S.(P. V.)Ex67.70, 68.70, 67.89, 68.904.600.960.44(<0.01)OLEx49.12, 50.14, 49.33, 49.341.30.220.85(<0.001)MEx54.32, 55.31, 54.54, 54.502.760.530.32(0.1)LBHEx67.70, 68.70, 67.89, 67.900.620.1050.44(<0.001)MOEx43.51, 45.51, 44.22, 43.900.80.140.18(0.55)GMOEx42.75, 45.74, 44.25, 43.340.510.080.15(0.78)KwEx41.78, 44.75, 43.28, 42.320.450.070.14(0.86)BEx43.48, 46.45, 44.98, 44.020.700.120.16(0.80)MOKEx41.58, 45.54, 44.25, 42.300.60,0.110.14(0.87)KMOEx42.82, 46.84, 45.55, 43.601.080.190.15(0.86)BrXEx48.13, 50.15, 48.83, 48.521.390.240.248(0.171)OIPGWEx38.74, 42.7, 41.41, 39.520.1780.03170.0999(0.9884)Figure 7
The TTT plot (a), Q-Q plot (b), and box plot (c) for the first data.
(a)(b)(c)Figure 8
E-CDF, E-PDF, and E-HRF for the first data.Figure 9
P-P plot and KM survival plot the first data.Based on Table13, we have come to the conclusion that the OIPGWEx model is much better than the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwE, BXEx, and KwMOEx models with C1 = 38.74, C2 = 42.7, C3 = 41.41, C4 = 39.52, A·=0.178, C·=0.0317, K. S = 0.0999, and P. V. = 0.9884, so the OIPGWEx model is an excellent alternative to these competitive distributions in modeling relief times data set. According to Figures 8 and 9, the OIPGWEx distribution provides adequate fits to the empirical functions.


### 8.2. Modeling Survival times

The survival data are the second data set. Recent analysis of these data was conducted by [29]. Table 14 lists the MLEs. Table 15 lists the C1, C2, C3, C4, A·, C·, K. S., and P. V. Figure 10 gives the TTT plot along with the corresponding box plot for the survival time data.Table 14
MLEs values for the second data.
ModelsMLEsExβ0.540OLExβ0.3815MExβ0.925LBHExβ0.54MOExα,β8.78, 1.38BrXExa,β0.48, 0.21GMOExλ,α,β0.18, 47.64, 4.47KwExa,θ,β3.304, 1.100, 1.037BExα,θ,β0.807, 3.461, 1.331MOKwExα,β,λ,β0.01, 2.72, 1.99, 0.01KwMOExα,β,λ,β0.37, 3.48, 3.31, 0.29OIPGWExλ,α,θ,β1.146, 0.26, 4.07, 4.5Table 15
C1, C2, C3, C4, A·, C·, K. S., and P. V. for second data.
ModelsC1, C2, C3, C4A·C·K. S.(P. V.)Ex234.60, 236.90, 234.68, 235.556.531.250.3(0.06)OLEx229.13, 231.43, 229.21, 230.111.940.330.5(<0.001)ME210.40, 212.68, 210.45, 211.301.520.250.15(0.13)LBHEx234.63, 236.92, 234.71, 235.510.710.1150.28(<0.001)MOEx210.37, 214.93, 210.52, 212.171.180.170.10(0.43)GMOEx210.54, 217.38, 210.89, 213.241.020.160.09(0.5)KwEx209.42, 216.24, 209.77, 212.120.740.110.09(0.5)BEx207.37, 214.21, 207.73, 210.09,0.980.150.11(0.34)MOKwEx209.44, 218.56, 210.04, 213.04,0.790.120.10(0.44)KwMOEx207.82, 216.94, 208.42, 211.420.610.110.09(0.5)BrXEx235.31, 239.92, 235.53, 237.142.920.520.22(0.002)OIPGWEx206.88, 215.98, 207.47, 210.500.630.1050.088(0.639)Figure 10
The TTT plot (a), Q-Q plot (b), and box plot (c) for the second data.
(a)(b)(c)Based on Figure10, the HRF of the survival times is “increasing HRF,” and these data have only four EV observations. Figure 11 gives the estimated PDF (E-PDF), E-CDF, E-HRF, and P-P plot survival time data. Figure 12 gives the Kaplan–Meier survival plot survival times data. Based on Table 15, we have come to the conclusion that the OIPGWEx model is far superior to the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwEx, BXEx, and KwMOEx models with C1 = 206.88, C2 = 215.98, C3 = 207.47, C4 = 210.50, A·=0.63, C·=0.105, K. S. = 0.088, and P. V. = 0.639, so the OIPGWEx model is an excellent alternative to these competitive distributions in modeling relief times data set. According to Figures 11 and 12, the OIPGWEx distribution provides adequate fits to the empirical functions.Figure 11
E-CDF, E-PDF, and E-HRF for second data.Figure 12
P-P plot and Kaplan–Meier survival plot for second data.


## 8.1. Modeling Failure (Relief) Times

The failure time data are the first data set. Recent analyses of these data were conducted by [27]. Table 12 lists the MLEs. Table 13 lists C1, C2, C3, C4, A·, C·, K. S., and P. V. The total time test (TTT) plot for the relief time data, as well as the related box plot, is shown in Figure 7. Based on Figure 7, the HRF of the relief times is “increasing HRF,” and these data have only one EV observation. Figure 8 gives the E-PDF, E-CDF, E-HRF, and P-P plot for relief time data. Figure 9 gives Kaplan–Meier (KM) survival plot for relief time data.Table 12
MLEs values for the first data.
ModelsMLEsExβ0.526OLExβ0.6044MExβ0.950LBHExβ0.5263MOExα,β54.474, 2.316BrXExa,β1.1635, 0.3207GMOExλ,α,β0.519, 89.462, 3.169KwExα,θ,β83.756, 0.568, 3.330BExα,θ,β81.633, 0.542, 3.514MOKwExλ,α,θ,β0.13, 33.24, 0.57, 1.67KwMOExλ,α,θ,β8.87, 34.83, 0.29, 4.89OIPGWExλ,α,θ,β1.971, 2.36, 0.51, 0.47Table 13
C1, C2, C3, C4, A·, C·, K. S., and P. V. for the first data.
ModelsC1, C2, C3, C4A·C·K. S.(P. V.)Ex67.70, 68.70, 67.89, 68.904.600.960.44(<0.01)OLEx49.12, 50.14, 49.33, 49.341.30.220.85(<0.001)MEx54.32, 55.31, 54.54, 54.502.760.530.32(0.1)LBHEx67.70, 68.70, 67.89, 67.900.620.1050.44(<0.001)MOEx43.51, 45.51, 44.22, 43.900.80.140.18(0.55)GMOEx42.75, 45.74, 44.25, 43.340.510.080.15(0.78)KwEx41.78, 44.75, 43.28, 42.320.450.070.14(0.86)BEx43.48, 46.45, 44.98, 44.020.700.120.16(0.80)MOKEx41.58, 45.54, 44.25, 42.300.60,0.110.14(0.87)KMOEx42.82, 46.84, 45.55, 43.601.080.190.15(0.86)BrXEx48.13, 50.15, 48.83, 48.521.390.240.248(0.171)OIPGWEx38.74, 42.7, 41.41, 39.520.1780.03170.0999(0.9884)Figure 7
The TTT plot (a), Q-Q plot (b), and box plot (c) for the first data.
(a)(b)(c)Figure 8
E-CDF, E-PDF, and E-HRF for the first data.Figure 9
P-P plot and KM survival plot the first data.Based on Table13, we have come to the conclusion that the OIPGWEx model is much better than the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwE, BXEx, and KwMOEx models with C1 = 38.74, C2 = 42.7, C3 = 41.41, C4 = 39.52, A·=0.178, C·=0.0317, K. S = 0.0999, and P. V. = 0.9884, so the OIPGWEx model is an excellent alternative to these competitive distributions in modeling relief times data set. According to Figures 8 and 9, the OIPGWEx distribution provides adequate fits to the empirical functions.


## 8.2. Modeling Survival times

The survival data are the second data set. Recent analysis of these data was conducted by [29]. Table 14 lists the MLEs. Table 15 lists the C1, C2, C3, C4, A·, C·, K. S., and P. V. Figure 10 gives the TTT plot along with the corresponding box plot for the survival time data.Table 14
MLEs values for the second data.
ModelsMLEsExβ0.540OLExβ0.3815MExβ0.925LBHExβ0.54MOExα,β8.78, 1.38BrXExa,β0.48, 0.21GMOExλ,α,β0.18, 47.64, 4.47KwExa,θ,β3.304, 1.100, 1.037BExα,θ,β0.807, 3.461, 1.331MOKwExα,β,λ,β0.01, 2.72, 1.99, 0.01KwMOExα,β,λ,β0.37, 3.48, 3.31, 0.29OIPGWExλ,α,θ,β1.146, 0.26, 4.07, 4.5Table 15
C1, C2, C3, C4, A·, C·, K. S., and P. V. for second data.
ModelsC1, C2, C3, C4A·C·K. S.(P. V.)Ex234.60, 236.90, 234.68, 235.556.531.250.3(0.06)OLEx229.13, 231.43, 229.21, 230.111.940.330.5(<0.001)ME210.40, 212.68, 210.45, 211.301.520.250.15(0.13)LBHEx234.63, 236.92, 234.71, 235.510.710.1150.28(<0.001)MOEx210.37, 214.93, 210.52, 212.171.180.170.10(0.43)GMOEx210.54, 217.38, 210.89, 213.241.020.160.09(0.5)KwEx209.42, 216.24, 209.77, 212.120.740.110.09(0.5)BEx207.37, 214.21, 207.73, 210.09,0.980.150.11(0.34)MOKwEx209.44, 218.56, 210.04, 213.04,0.790.120.10(0.44)KwMOEx207.82, 216.94, 208.42, 211.420.610.110.09(0.5)BrXEx235.31, 239.92, 235.53, 237.142.920.520.22(0.002)OIPGWEx206.88, 215.98, 207.47, 210.500.630.1050.088(0.639)Figure 10
The TTT plot (a), Q-Q plot (b), and box plot (c) for the second data.
(a)(b)(c)Based on Figure10, the HRF of the survival times is “increasing HRF,” and these data have only four EV observations. Figure 11 gives the estimated PDF (E-PDF), E-CDF, E-HRF, and P-P plot survival time data. Figure 12 gives the Kaplan–Meier survival plot survival times data. Based on Table 15, we have come to the conclusion that the OIPGWEx model is far superior to the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwEx, BXEx, and KwMOEx models with C1 = 206.88, C2 = 215.98, C3 = 207.47, C4 = 210.50, A·=0.63, C·=0.105, K. S. = 0.088, and P. V. = 0.639, so the OIPGWEx model is an excellent alternative to these competitive distributions in modeling relief times data set. According to Figures 11 and 12, the OIPGWEx distribution provides adequate fits to the empirical functions.Figure 11
E-CDF, E-PDF, and E-HRF for second data.Figure 12
P-P plot and Kaplan–Meier survival plot for second data.


## 9. Conclusions

A novel flexibly generated family of distributions was developed and investigated in this study which is called OIPGW-G family. The new proposed family contains many new models, and its density can be right skewed and symmetric with unimodal and bimodal shapes. The new HRF of the new models can be “constant,” “decreasing,” “increasing,” “increasing-constant,” “upside-down-constant,” and “decreasing-constant.” Some of the mathematical properties of the new family are computed. Numerical calculations for the expected value, skewness, variance, and kurtosis are computed. Different types of entropies are calculated. Some numerical values of RéE, QE, HaChE, and ArE for some selected values of parameters for the OIPGWEx model are computed. Estimation of OIPGW-G parameters is performed by ML, LS, and WLS estimation methods. Some bivariate and multivariate OIPGWEx type models have been also derived. Two genuine data sets are used to demonstrate the family’s utility and versatility. The OIPGWEx model is far superior to the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwEx, BXEx, and KwMOEx models in modeling the two data sets according to theC1, C2, C3, C4, A·, C·, K. S., and P. V. statistics. In the future, we are planning to use this family to generate new statistical models and study its structural properties.



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*Source: 5082192-2021-11-09.xml*