# The Extended Inverse Weibull Distribution: Properties and Applications

Authors: Said Alkarni; Ahmed Z. Afify; I. Elbatal; M. Elgarhy
Journal: Complexity
 (2020)

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## Abstract

This paper proposes the new three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution which generalizes the inverse Weibull model. The density function of the TIHLIW can be expressed as a linear combination of the inverse Weibull densities. Some mathematical quantities of the proposed TIHLIW model are derived. Four estimation methods, namely, the maximum likelihood, least squares, weighted least squares, and Cramér–von Mises methods, are utilized to estimate the TIHLIW parameters. Simulation results are presented to assess the performance of the proposed estimation methods. The importance of the TIHLIW model is studied via a real data application.


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## Body

## 1. Introduction

The inverse Weibull (IW) distribution is also known as reciprocal Weibull distribution (see [1, 2]). Keller et al. [3] used the IW distribution to describe the degradation phenomena of mechanical components such as crankshaft and pistons of diesel engines. Further, the IW model has many important applications in reliability engineering, infant mortality, useful life, wear-out periods, life testing, and service records (see [4]).The cumulative distribution function (CDF) of the IW model is(1)GIWx;α,β=e−αx−β,x>0,α,β>0.Its associated probability density function (PDF) has the following form:(2)gIWx;α,β=αβx−β−1e−αx−β,x>0,α,β>0.The statistical literature contains several extensions of the IW model, see, for example, beta IW by Khan [5], generalized IW by de Gusmão et al. [6], modified IW by Khan and King [4], gamma IW by Pararai et al. [7], Kumaraswamy generalized IW by Oluyede and Yang [8], Kumaraswamy modified IW by Aryal and Elbatal [9], Marshall-Olkin IW by Okasha et al. [10], and alpha power IW by Basheer [11].Many generalized classes of distributions have been proposed for modeling real-life data in several applied fields such as reliability, engineering, biological studies, economics, medical sciences, environmental sciences, and finance. For example, Marshall and Olkin [12] proposed Marshall-Olkin-G, Shaw and Buckley [13] defined transmuted-G, Cordeiro and de Castro [14] pioneered Kumaraswamy-G, Alizadeh et al. [15] proposed the extended odd Weibull-G, and Cordeiro et al. [16] studied the odd Lomax-G family.Cordeiro et al. [17] proposed the type I half-logistic-G (TIHL-G) class. The CDF of the TIHL-G family of distributions is given (for x∈ℝ) by(3)Fx;λ,δ=1−1−Gx;δλ1+1−Gx;δλ,λ>0,where Gx;δ refers to the baseline CDF with a parameter vector δ. The CDF in (3) is a wider class which can be used to generate more flexible extended distributions.The associated PDF of (3) has the form(4)fx;λ,δ=2λgx;δ1−Gx;δλ−11+1−Gx;δλ2,where gx;δ is the baseline PDF. The hazard rate function (HRF) of the TIHL-G family is(5)hx,λ,δ=λgx;δ1−Gx;δ1+1−Gx;δλ.In this paper, we propose a new lifetime model called the type I half-logistic inverse Weibull (TIHLIW) model. The proposed model can be used, as a good alternative to some existing distributions, in modeling several real data.The paper is outlined as follows. In Section2, we define the TIHLIW distribution and derive a useful representation for its PDF. The mathematical properties of the TIHLIW distribution are derived in Section 3. In Section 4, the TIHLIW parameters are estimated via four methods, namely, the maximum likelihood, least squares, weighted least squares, and Cramér–von Mises estimators. These estimators are compared via some simulations in Section 5. In Section 6, we illustrate the flexibility and potentiality of the TIHLIW model using a real data set. Finally, some concluding remarks are offered in Section 7.


## 2. The TIHLIW Distribution

The CDF of the three-parameter TIHLIW distribution follows, by replacing equation (1) in (3), as(6)Fx;λ,α,β=1−1−e−αx−βλ1+1−e−αx−βλ.The corresponding PDF of (4) reduces to(7)fx;λ,α,β=2λαβx−β−1e−αx−β1−e−αx−βλ−11+1−e−αx−βλ2.The HRF of the TIHLIW distribution takes the form(8)hx;λ,α,β=λαβx−β−1e−αx−β1−e−αx−β1+1−e−αx−βλ.Figure1 provides some shapes of the PDF and HRF of the TIHLIW distribution for some different values of the parameters.Figure 1
Plots of the PDF and HRF of the TIHLIW distribution for different values of parameters.The TIHLIW distribution is a very flexible model that approaches to different distributions as special submodels:(1)
Ifα=1, then the TIHLIW distribution reduces to the type I half-logistic Fréchet (TIHLFr) distribution(2)
Ifβ=1, TIHLIW distribution reduces to the type I half-logistic inverse exponential (TIHLIE) distribution(3)
Ifβ=2, we have the type I half-logistic inverse Rayleigh (TIHLIR) distribution


### 2.1. Linear Representation

In this section, we express the TIHLIW PDF as a mixture linear representation of IW densities.Consider the power series:(9)1+z−k=∑n=0∞−knzn,z<1andk>0.Expanding1+1−e−αx−βλ−2 using (9), we can write (7) as(10)fx;λ,α,β=2λαβx−β−1e−αx−β∑i=0∞−2i1−e−αx−βλi+1−1.Consider the power series:(11)1−zb−1=∑j=0∞−1jb−1jzj,z<1andb>0.Using the power series (11) and after some algebra, the TIHLIW PDF reduces to(12)fx;λ,α,β=2λαβx−β−1∑i,j=0∞−1j−2iλi+1−1je−αj+1x−β=∑j=0∞φjgx;αj+1,β,where gx;αj+1,β refers to the IW PDF with parameters αj+1 and β, and(13)φj=2λ∑i=0∞−1jj+1−2iλi+1−1j.Equation (12) means that the PDF of the TIHLIW is a linear combination of the IW densities and can be used to calculate some mathematical quantities of the TIHLIW model from those of the IW distribution.Consider the random variableRVY∼IWα,β in (1). For n<β, the nth ordinary and incomplete moments of Y are given by(14)μn,Y′=αn/βΓ1−nβ,φn,Yt=αn/βγ1−nβ,αtβ,respectively, where Γb=∫0∞wb−1e−wdw is the complete gamma function and γb,z=∫0zwb−1e−wdw is the lower incomplete gamma function.


## 2.1. Linear Representation

In this section, we express the TIHLIW PDF as a mixture linear representation of IW densities.Consider the power series:(9)1+z−k=∑n=0∞−knzn,z<1andk>0.Expanding1+1−e−αx−βλ−2 using (9), we can write (7) as(10)fx;λ,α,β=2λαβx−β−1e−αx−β∑i=0∞−2i1−e−αx−βλi+1−1.Consider the power series:(11)1−zb−1=∑j=0∞−1jb−1jzj,z<1andb>0.Using the power series (11) and after some algebra, the TIHLIW PDF reduces to(12)fx;λ,α,β=2λαβx−β−1∑i,j=0∞−1j−2iλi+1−1je−αj+1x−β=∑j=0∞φjgx;αj+1,β,where gx;αj+1,β refers to the IW PDF with parameters αj+1 and β, and(13)φj=2λ∑i=0∞−1jj+1−2iλi+1−1j.Equation (12) means that the PDF of the TIHLIW is a linear combination of the IW densities and can be used to calculate some mathematical quantities of the TIHLIW model from those of the IW distribution.Consider the random variableRVY∼IWα,β in (1). For n<β, the nth ordinary and incomplete moments of Y are given by(14)μn,Y′=αn/βΓ1−nβ,φn,Yt=αn/βγ1−nβ,αtβ,respectively, where Γb=∫0∞wb−1e−wdw is the complete gamma function and γb,z=∫0zwb−1e−wdw is the lower incomplete gamma function.


## 3. Some Properties

In this section, we studied some statistical properties of the TIHLIW distribution, such as quantile function, ordinary moments, moment generating function, incomplete moment, and mean deviation.


### 3.1. Quantile and Moment-Generating Functions

AsRVX has CDF of TIHLIW distribution, the quantile function (QF) is defined by(15)Qu=infx∈R:Fx≥u,where 0<u<1. This relation is used to find the QF of TIHLIW distribution as follows:(16)Qu=−1αln1−1−u1+uλ−1/β,u∈0,1.The above equation can be used to generate TIHLIW random variates. Here, we obtain the MGF of the IW distribution (1) by setting w=x−1:(17)Mt;α,β=αβ∫0∞exptwwβ−1exp−αwβdw.By expandingexpt/w and calculating the integral, we can write(18)Mt;α,β=αβ∫0∞∑n=0∞tnn!wβ−n−1exp−αwβdw=∑n=0∞αn/βtnn!Γβ−nβ.Using the Wright generalized hypergeometric function [18],(19)ψpqα1,C1,…,αp,Cpβ1,D1,…,βq,Dq;x=∑m=0∞∏i=1pΓαi+Cim∏i=1qΓβi+Dimxmm!.The MGF of the IW distribution has the form(20)Mt;α,β=ψ101,−β−1−;α1/βt.Using equations (12) and (20), the MGF of the TIHLIW distribution reduces to(21)Mt=∑j=0∞φj1ψ01,−β−1−;αj+11/βt.


### 3.2. Moments

Thesth ordinary moment of RVX is(22)μs′=EXs=∑j=0∞φj∫0∞xsgx;αj+1,βdx.Fors<β, we obtain(23)μs′=∑j=0∞φjαj+1s/βΓ1−sβ.Settings=1 in (23), we obtain the mean of X.Thesth incomplete moment of the TIHLIW distribution is defined by φst=∫0txsfxdx.Using equation (12), we can write(24)φst=∑j=0∞φj∫0txsgx;αj+1,β.Hence, we obtain thesth incomplete moment of the TIHLIW distribution:(25)φst=∑j=0∞φjαj+1s/βγ1−sβ,αj+1tβ,s<β.The first incomplete moment which follows by settings=1 in the above equation is(26)φ1t=∑j=0∞φjαj+11/βγ1−1β,αj+1tβ,β>1.


### 3.3. Mean Residual Life and Mean Waiting Time

The mean residual life (MRL) has useful applications in economics, life insurance, biomedical sciences, demography, product quality control, and product technology (see [19]). The MRL refers to the expected additional life length for a unit that is alive at age t, and it is defined by mXt=EX−tX>t, t>0.The MRL ofX can be calculated by the formula:(27)mXt=1−φ1tSt−t,where St is the survival function of X.By inserting (26) in (27), the MRL of the TIHLIW distribution follows as(28)mXt=1St∑j=0∞φjαj+11/βγ1−1β,αj+1tβ−t.The mean inactivity time (MIT) (mean waiting time) is defined byMXt=Et−XX≤t, t>0, and it can be calculated by the formula:(29)MXt=t−φ1tFt.By substituting (26) in (29), the MIT of the TIHLIW distribution follows as(30)MXt=t−1Ft∑j=0∞φjαj+11/βγ1−1β,αj+1tβ.


### 3.4. Order Statistics

Consider the random sample from the TIHLIWλ,α,β denoted by X1,…,Xn and its associated order statistics denoted by X1,…,Xn. The PDF of the rth order statistic is denoted by Xr,1≤r≤n and can be expressed as(31)frx=n!r−1!n−r!2λαβx−β−1e−αx−β1−e−αx−βλ−11+1−e−αx−βλ2×1−1−e−αx−βλ1+1−e−αx−βλr−121−e−αx−βλ1+1−e−αx−βλn−r.The PDF ofXr is defined also by the formula:(32)frx=fxBr,n−r+1∑j=0n−r−1jn−rjFj+r−1x.Using the PDF and CDF of the TIHLIW distribution, equation (32) reduces to(33)fxFj+r−1x=2λαβx−β−1e−αx−β1−e−αx−βλ−11+1−e−αx−βλj+r+11−1−e−αx−βλj+r−1.Using expansion (9) and the power series,(34)1−zb−1=∑j=0b−1−1jb−1jzj,bpositive integer.We can write(35)fxFj+r−1x=2λαβx−β−1e−αx−β∑l=0∞∑i=0j+r−1−1i−j−r−1l×j+r−1i1−e−αx−βλl+i+1−1.Applying the power series (11), equation (35) reduces to(36)fxFj+r−1x=2λαβx−β−1∑l=0∞∑i=0j+r−1−1i−j−r−1lj+r−1i×λl+i+1−1ke−αk+1x−βBy replacing (36) in equation (32), we obtain(37)frx=2λBr,n−r+1∑j=0n−r∑l,k=0∞∑i=0j+r−1−1j+i+kn−rj−j−r−1l×j+r−1iλl+i+1−1kαβx−β−1e−αk+1x−β.Hence, we can write(38)frx=∑k=0∞bkgx;αk+1,β,where gx;αk+1,β denotes the IW PDF with parameters β and αk+1, and(39)bk=∑j=0n−r∑l=0∞∑i=0j+r−12λ−1j+i+kBr,n−r+1k+1n−rj−j−r−1lj+r−1iλl+i+1−1k.Equation (38) reveals that the PDF of the TIHLIW order statistics is a mixture of IW densities. Hence, the qth moment of Xr has the form(40)EXrq=∑k=0∞bkαk+1q/βΓ1−qβ.


## 3.1. Quantile and Moment-Generating Functions

AsRVX has CDF of TIHLIW distribution, the quantile function (QF) is defined by(15)Qu=infx∈R:Fx≥u,where 0<u<1. This relation is used to find the QF of TIHLIW distribution as follows:(16)Qu=−1αln1−1−u1+uλ−1/β,u∈0,1.The above equation can be used to generate TIHLIW random variates. Here, we obtain the MGF of the IW distribution (1) by setting w=x−1:(17)Mt;α,β=αβ∫0∞exptwwβ−1exp−αwβdw.By expandingexpt/w and calculating the integral, we can write(18)Mt;α,β=αβ∫0∞∑n=0∞tnn!wβ−n−1exp−αwβdw=∑n=0∞αn/βtnn!Γβ−nβ.Using the Wright generalized hypergeometric function [18],(19)ψpqα1,C1,…,αp,Cpβ1,D1,…,βq,Dq;x=∑m=0∞∏i=1pΓαi+Cim∏i=1qΓβi+Dimxmm!.The MGF of the IW distribution has the form(20)Mt;α,β=ψ101,−β−1−;α1/βt.Using equations (12) and (20), the MGF of the TIHLIW distribution reduces to(21)Mt=∑j=0∞φj1ψ01,−β−1−;αj+11/βt.


## 3.2. Moments

Thesth ordinary moment of RVX is(22)μs′=EXs=∑j=0∞φj∫0∞xsgx;αj+1,βdx.Fors<β, we obtain(23)μs′=∑j=0∞φjαj+1s/βΓ1−sβ.Settings=1 in (23), we obtain the mean of X.Thesth incomplete moment of the TIHLIW distribution is defined by φst=∫0txsfxdx.Using equation (12), we can write(24)φst=∑j=0∞φj∫0txsgx;αj+1,β.Hence, we obtain thesth incomplete moment of the TIHLIW distribution:(25)φst=∑j=0∞φjαj+1s/βγ1−sβ,αj+1tβ,s<β.The first incomplete moment which follows by settings=1 in the above equation is(26)φ1t=∑j=0∞φjαj+11/βγ1−1β,αj+1tβ,β>1.


## 3.3. Mean Residual Life and Mean Waiting Time

The mean residual life (MRL) has useful applications in economics, life insurance, biomedical sciences, demography, product quality control, and product technology (see [19]). The MRL refers to the expected additional life length for a unit that is alive at age t, and it is defined by mXt=EX−tX>t, t>0.The MRL ofX can be calculated by the formula:(27)mXt=1−φ1tSt−t,where St is the survival function of X.By inserting (26) in (27), the MRL of the TIHLIW distribution follows as(28)mXt=1St∑j=0∞φjαj+11/βγ1−1β,αj+1tβ−t.The mean inactivity time (MIT) (mean waiting time) is defined byMXt=Et−XX≤t, t>0, and it can be calculated by the formula:(29)MXt=t−φ1tFt.By substituting (26) in (29), the MIT of the TIHLIW distribution follows as(30)MXt=t−1Ft∑j=0∞φjαj+11/βγ1−1β,αj+1tβ.


## 3.4. Order Statistics

Consider the random sample from the TIHLIWλ,α,β denoted by X1,…,Xn and its associated order statistics denoted by X1,…,Xn. The PDF of the rth order statistic is denoted by Xr,1≤r≤n and can be expressed as(31)frx=n!r−1!n−r!2λαβx−β−1e−αx−β1−e−αx−βλ−11+1−e−αx−βλ2×1−1−e−αx−βλ1+1−e−αx−βλr−121−e−αx−βλ1+1−e−αx−βλn−r.The PDF ofXr is defined also by the formula:(32)frx=fxBr,n−r+1∑j=0n−r−1jn−rjFj+r−1x.Using the PDF and CDF of the TIHLIW distribution, equation (32) reduces to(33)fxFj+r−1x=2λαβx−β−1e−αx−β1−e−αx−βλ−11+1−e−αx−βλj+r+11−1−e−αx−βλj+r−1.Using expansion (9) and the power series,(34)1−zb−1=∑j=0b−1−1jb−1jzj,bpositive integer.We can write(35)fxFj+r−1x=2λαβx−β−1e−αx−β∑l=0∞∑i=0j+r−1−1i−j−r−1l×j+r−1i1−e−αx−βλl+i+1−1.Applying the power series (11), equation (35) reduces to(36)fxFj+r−1x=2λαβx−β−1∑l=0∞∑i=0j+r−1−1i−j−r−1lj+r−1i×λl+i+1−1ke−αk+1x−βBy replacing (36) in equation (32), we obtain(37)frx=2λBr,n−r+1∑j=0n−r∑l,k=0∞∑i=0j+r−1−1j+i+kn−rj−j−r−1l×j+r−1iλl+i+1−1kαβx−β−1e−αk+1x−β.Hence, we can write(38)frx=∑k=0∞bkgx;αk+1,β,where gx;αk+1,β denotes the IW PDF with parameters β and αk+1, and(39)bk=∑j=0n−r∑l=0∞∑i=0j+r−12λ−1j+i+kBr,n−r+1k+1n−rj−j−r−1lj+r−1iλl+i+1−1k.Equation (38) reveals that the PDF of the TIHLIW order statistics is a mixture of IW densities. Hence, the qth moment of Xr has the form(40)EXrq=∑k=0∞bkαk+1q/βΓ1−qβ.


## 4. Estimation Methods

The estimation of the TIHLIW parameters is investigated using four methods of estimation, namely, the maximum likelihood estimators (MLEs), least squares estimators (LSEs), weighted least squares estimators (WLSEs), and Cramér–von Mises estimators (CVMEs).


### 4.1. Maximum Likelihood

We determine the MLEs of the TIHLIW parameters. Letx1,x2,…,xn be a random sample of size n from TIHLIW ϕ where ϕ=λ,α,βT. The log-likelihood function for ϕ has the form(41)ℓ=nlog2λαβ−β+1∑i=1nlogxi−α∑i=1nxiβ+λ−1∑i=1nlog1−e−αxi−β−2∑i=1nlog1+1−e−αxi−β.We can maximize the above log-likelihood equation by solving the nonlinear likelihood equations which follow by differentiating it. The associated components of the score vector(42)Unϕ=∂ℓ∂λ,∂ℓ∂α,∂ℓ∂βT,are given by(43)∂ℓ∂λ=nλ+∑i=1nlog1−e−αxi−β,∂ℓ∂α=nα−∑i=1nxiβ+λ−1∑i=1nxi−βe−αxi−β1−e−αxi−β−2∑i=1nxi−βe−αxi−β1+1−e−αxi−β.And,(44)∂ℓ∂β=nβ−∑i=1nlogxi−α∑i=1nxiβlogxi+2∑i=1nαlogxixi−βe−αxi−β1+1−e−αxi−β+λ−1∑i=1nαlogxixi−βe−αxi−β1−e−αxi−β.The MLEs ofϕ can be constructed by solving the nonlinear system Unϕ=0 that cannot be analytically solved; hence, statistical programs are utilized to solve them numerically via iterative techniques such as a Newton–Raphson algorithm.


### 4.2. Ordinary and Weighted Least Squares

The least square and weighted least square methods are used to estimate the parameters of beta distribution [20]. Let x1<x2<⋯<xn be the order statistics of a sample from the TIHLIW distribution, and then the LSEs and WLSEs of α,λ, and β can be obtained by minimizing the following function with respect to α,λ, and β:(45)Sα,λ,β=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+12,where Ai=1 in the case of LSEs and Ai=n+12n+2/in−i+1 in the case of WLSEs.Further, the LSEs and WLSEs of the TIHLIW parameters are also obtained by solving the following nonlinear equations simultaneously with respect toα,λ, and β:(46)∂Sα,λ,β∂α=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ1i=0,∂Sα,λ,β∂λ=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ2i=0,∂Sα,λ,β∂β=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ3i=0,where(47)ϕ1i=−2λxi−βe−αxi−β1−e−αxi−βλ−11+1−e−αxi−βλ2,(48)ϕ2i=21−e−αxi−β2λlog1−e−αxi−β1+1−e−αxi−βλ2,(49)ϕ3i=2λαxi−βe−αxi−βlogxi1−e−αxi−βλ−11+1−e−αxi−βλ2.


### 4.3. Cramér–von Mises

Based on the Cramér–von Mises (CVM) method [21, 22], the CVMEs of the TIHLIW parameters can be constructed by minimizing(50)Cα,λ,β=112n+∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12n2,with respect to α,λ, and β. Further, the CVMEs can also be obtained by solving the following nonlinear equations simultaneously with respect to α,λ, and β:(51)∂Cα,λ,β∂α=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ1i=0,∂Cα,λ,β∂λ=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ2i=0,∂Cα,λ,β∂β=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ3i=0,where ϕ1i, ϕ2i, and ϕ3i are, respectively, defined in equations (47)–(49).


## 4.1. Maximum Likelihood

We determine the MLEs of the TIHLIW parameters. Letx1,x2,…,xn be a random sample of size n from TIHLIW ϕ where ϕ=λ,α,βT. The log-likelihood function for ϕ has the form(41)ℓ=nlog2λαβ−β+1∑i=1nlogxi−α∑i=1nxiβ+λ−1∑i=1nlog1−e−αxi−β−2∑i=1nlog1+1−e−αxi−β.We can maximize the above log-likelihood equation by solving the nonlinear likelihood equations which follow by differentiating it. The associated components of the score vector(42)Unϕ=∂ℓ∂λ,∂ℓ∂α,∂ℓ∂βT,are given by(43)∂ℓ∂λ=nλ+∑i=1nlog1−e−αxi−β,∂ℓ∂α=nα−∑i=1nxiβ+λ−1∑i=1nxi−βe−αxi−β1−e−αxi−β−2∑i=1nxi−βe−αxi−β1+1−e−αxi−β.And,(44)∂ℓ∂β=nβ−∑i=1nlogxi−α∑i=1nxiβlogxi+2∑i=1nαlogxixi−βe−αxi−β1+1−e−αxi−β+λ−1∑i=1nαlogxixi−βe−αxi−β1−e−αxi−β.The MLEs ofϕ can be constructed by solving the nonlinear system Unϕ=0 that cannot be analytically solved; hence, statistical programs are utilized to solve them numerically via iterative techniques such as a Newton–Raphson algorithm.


## 4.2. Ordinary and Weighted Least Squares

The least square and weighted least square methods are used to estimate the parameters of beta distribution [20]. Let x1<x2<⋯<xn be the order statistics of a sample from the TIHLIW distribution, and then the LSEs and WLSEs of α,λ, and β can be obtained by minimizing the following function with respect to α,λ, and β:(45)Sα,λ,β=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+12,where Ai=1 in the case of LSEs and Ai=n+12n+2/in−i+1 in the case of WLSEs.Further, the LSEs and WLSEs of the TIHLIW parameters are also obtained by solving the following nonlinear equations simultaneously with respect toα,λ, and β:(46)∂Sα,λ,β∂α=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ1i=0,∂Sα,λ,β∂λ=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ2i=0,∂Sα,λ,β∂β=∑i=1nAi1−1−e−αxi−βλ1+1−e−αxi−βλ−in+1ϕ3i=0,where(47)ϕ1i=−2λxi−βe−αxi−β1−e−αxi−βλ−11+1−e−αxi−βλ2,(48)ϕ2i=21−e−αxi−β2λlog1−e−αxi−β1+1−e−αxi−βλ2,(49)ϕ3i=2λαxi−βe−αxi−βlogxi1−e−αxi−βλ−11+1−e−αxi−βλ2.


## 4.3. Cramér–von Mises

Based on the Cramér–von Mises (CVM) method [21, 22], the CVMEs of the TIHLIW parameters can be constructed by minimizing(50)Cα,λ,β=112n+∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12n2,with respect to α,λ, and β. Further, the CVMEs can also be obtained by solving the following nonlinear equations simultaneously with respect to α,λ, and β:(51)∂Cα,λ,β∂α=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ1i=0,∂Cα,λ,β∂λ=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ2i=0,∂Cα,λ,β∂β=∑i=1n1−1−e−αxi−βλ1+1−e−αxi−βλ−2i−12nϕ3i=0,where ϕ1i, ϕ2i, and ϕ3i are, respectively, defined in equations (47)–(49).


## 5. Simulation Study

In this section, we conduct a simulation study to compare the performance of the different estimators based on the mean square error criterion. We compare the performances of the MLEs, LSEs, WLSEs, and CVMEs based on the mean square errors (MSEs) for different sample sizes. Mathematica 9 is utilized to obtain simulation results from10,000 replications. The mean values of the MLEs, LSEs, WLSEs, and CVMEs and their MSEs of the TIHLIW parameters by considering sample sizes n=50, 100, and 200, are given for different values of parameters α, λ, and β. The mean values of the MLEs, LSEs, WLSEs, and CVMEs and MSEs of the TIHLIW parameters are listed in Tables 1–3.Table 1
The mean values and MSEs of the TIHLIW distribution.
ParametersMLEsLSEsWLSEsCVMEsn=50α=0.50.559 (0.244)0.620 (0.223)0.612 (0.201)0.633 (0.240)λ=0.50.577 (0.269)0.654 (0.280)0.626 (0.189)0.659 (0.299)β=0.50.646 (0.122)0.540 (0.066)0.540 (0.062)0.567 (0.078)n=100α=0.50.522 (0.103)0.588 (0.117)0.578 (0.103)0.574 (0.128)λ=0.50.521 (0.061)0.593 (0.099)0.575 (0.071)0.58 (0.122)β=0.50.574 (0.048)0.521 (0.042)0.518 (0.033)0.543 (0.045)n=200α=0.50.504 (0.055)0.538 (0.072)0.532 (0.059)0.553 (0.072)λ=0.50.509 (0.029)0.546 (0.049)0.535 (0.034)0.555 (0.051)β=0.50.539 (0.022)0.522 (0.025)0.516 (0.018)0.515 (0.023)Table 2
The mean values and MSEs of the TIHLIW distribution.
ParametersMLEsLSEsWLSEsCVMEsn=50α=0.50.552 (0.211)0.474 (0.110)0.499 (0.114)0.466 (0.104)λ=1.51.877 (3.526)1.056 (1.525)0.985 (1.589)1.492 (0.800)β=0.50.612 (0.094)0.568 (0.033)0.565 (0.039)0.594 (0.042)n=100α=0.50.551 (0.119)0.456 (0.062)0.489 (0.07)0.454 (0.053)λ=1.51.737 (1.128)1.431 (0.453)1.523 (0.515)1.423 (0.403)β=0.50.542 (0.032)0.558 (0.019)0.544 (0.019)0.563 (0.019)n=200α=0.50.516 (0.051)0.433 (0.03)0.459 (0.031)0.437 (0.029)λ=1.51.578 (0.340)1.351 (0.179)1.419 (0.187)1.368 (0.175)β=0.50.524 (0.013)0.556 (0.012)0.540 (0.010)0.553 (0.011)Table 3
The mean values and MSEs of the TIHLIW distribution.
ParametersMLEsLSEsWLSEsCVMEsn=50α=0.50.602 (0.276)0.544 (0.060)0.545 (0.056)0.563 (0.064)λ=1.52.126 (5.703)1.656 (0.641)1.652 (0.469)1.704 (0.627)β=1.51.786 (0.764)1.498 (0.106)1.510 (0.116)1.508 (0.095)n=100α=0.50.555 (0.128)0.539 (0.021)0.552 (0.026)0.527 (0.017)λ=1.51.775 (1.598)1.600 (0.128)1.642 (0.161)1.583 (0.096)β=1.51.618 (0.297)1.452 (0.036)1.447 (0.047)1.493 (0.040)n=200α=0.50.516 (0.049)0.524 (0.009)0.540 (0.014)0.530 (0.009)λ=1.51.583 (0.346)1.562 (0.042)1.608 (0.082)1.577 (0.045)β=1.51.567 (0.113)1.470 (0.017)1.456 (0.025)1.467 (0.016)From Tables1–3, we conclude the following:(1)
The MSE ofα, λ, and β for all estimation methods decreases as n increases.(2)
Table1 shows that the MLEs have the lowest MSE in most cases of α and λ. Also, the WLSEs have the least MSE in most cases of β at α=0.5, λ=0.5, and β=0.5.(3)
Table2 shows that CVMEs get the least MSE in most situations of α and λ. Also, the WLSEs have the lowest MSE in most cases of β at α=0.5, λ=1.5, and β=0.5.(4)
Table3 shows that CVMEs have the lowest MSE in most cases of α, λ, and β at α=0.5, λ=1.5, and β=1.5.


## 6. Data Analysis

In this section, we present an application to a real data set to illustrate the performance and flexibility of the TIHLIW distribution. The data refer to relief times of a sample of 20 patients who receive an analgesic [23]. These data have been analyzed by Afify et al. [24] and Cordeiro et al. [25].For these data, we compare the TIHLIW model with some rival models, namely, the beta generalized inverse Weibull geometric (BGIWGc) by Elbatal et al. [26], transmuted complementary Weibull geometric (TCWGc) by Afify et al. [27], beta transmuted Weibull (BTW) by Afify et al. [28], McDonald log-logistic (McLL) by Tahir et al. [29], beta Weibull (BW) by Lee et al. [30], McDonald Weibull (McW) by Cordeiro et al. [31], exponentiated transmuted generalized Rayleigh (ETGR) by Afify et al. [32], and new modified Weibull (NMW) [33] distributions. The PDFs of these distributions are given (for x>0) byBGIWGc:fx=1−pb/Ba,bαθγαx−θ−1e−γaαx−θ1−e−γαx−θb−1×1−p1−e−γαx−θ−a−b, where p∈0,1, α,β,γ,a>0 and b>0.BTW:fx=βαβ/Ba,bxβ−11−λ+2λe−αxβ1−e−αxβ1+λe−αxβa−1×e−αxβ1−1−e−αxβ1+λe−αxβb−1, where α,β,a,b>0 and λ≤1.TCWGc:fx=αβγγyβ−1e−γyβα1−λ−α−αλ−λ−1e−γyβ×α+1−αe−γyβ−3, where α,β,γ>0 and λ≤1.McLL:fx=αc/Ba/c,bβx/βaα−11+x/βα−a−11−1−1+x/βα−1c, where α,β,a,b>0 and c>0.McW:fx=βcαβ/Ba/c,bxβ−1e−αxβ1−e−αxβa−11−1−e−αxβcb−1, where α,β,a,b>0 and c>0.BW:fx=βαβ/Ba,bxβ−1e−bαxβ1−e−αxβa−1, where α,β,a>0 and b>0.ETGR:fx=2αδβ2xe−βx21−e−βx2αδ−1×1+λ−2λ1−e−βx2α1+λ−λ1−e−βx2αα−1, where α,β,δ>0 and λ≤1.NMW:fx=αθxθ−1+γβ+δxxβ−1eδxe−αxθ+γxβeδx, where α,β,θ,γ>0 and δ>0.We consider some criteria including−2ℓ^ (where ℓ^ is the maximized log-likelihood), AIC (Akaike information criterion), HQIC (Hannan–Quinn information criterion), CAIC (corrected Akaike information Criterion), AD (Anderson–Darling statistic), and CVM (Cramér–von Mises statistic), where Table 4 lists the numerical values of −2ℓ^, AIC, CAIC, HQIC, A∗, and W∗ for all fitted models, whereas MLEs and their standard errors (SEs) (in parentheses) are given in Table 5. From Table 4, the TIHLIW has the lowest values for all goodness-of-fit measures, and hence it provides close fits to relief times data than other fitted models.Table 4
Goodness-of-fit measures for relief times data.
Model−2ℓ^AICCAICHQICADCVMTIHLIW30.75236.75238.25237.3350.17460.0319BGIWGc31.66243.66250.12444.8280.24670.0434BTW33.05143.05174.33744.0230.39770.0689TCWGc33.60741.60944.27442.3850.43600.0725McLL33.85443.85548.14044.8260.46200.0790McW33.90743.90748.19344.8790.46930.0802BW34.39642.39645.06343.1740.51320.0873ETGR36.85644.85747.52345.6340.79290.1363NMW41.17351.17355.45952.1451.06780.1759Table 5
MLEs and SEs for relief times data.
ModelMLEsTIHLIW5.3175.3610.799α,β,λ(2.583)(3.712)(0.789)BGIWGc19.1820.591.4349.84839⋅10−55.8015α,γ,θ,p,a,b(33.03)(43.24)(0.837)(2.001)(63.25)(4.346)BTW5.6180.53153.343.568−0.771α,β,a,b,λ(9.353)(0.184)(111.4)(4.265)(3.894)TCWGc43.665.1270.282−0.271α,β,γ,λ(45.45)(0.814)(0.042)(0.656)McLL0.8812.07019.2232.031.926α,β,a,b,c(0.109)(3.693)(22.34)(43.07)(5.165)McW2.7730.38079.1017.893.006α,β,a,b,c(6.380)(0.188)(119.1)(39.51)(13.96)BW0.8310.61229.9411.63α,β,a,b(0.954)(0.340)(40.41)(21.90)ETGR0.1030.691−0.34223.53α,β,λ,δ(0.436)(0.086)(1.971)(105.3)NMW0.1212.7838.2⋅10−50.00032.787α,β,γ,δ,θ(0.056)(20.37)(1.5⋅10−3 )(0.025)(0.428)The fitted PDF, estimated CDF, estimated survival function (SF), and PP plots of the TIHLIW distribution are shown in Figures2 and 3.Figure 2
The fitted PDF and estimated CDF of the TIHLIW model.Figure 3
The estimated SF and PP plot of the TIHLIW model.


## 7. Conclusions

We proposed a three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution as a new extension of the inverse Weibull model. The TIHLIW density is a linear combination of the inverse Weibull densities. Some explicit expressions for mathematical quantities of the TIHLIW distribution are derived. We consider four methods of estimation, namely, the maximum likelihood, least squares, weighted least squares, and Cramér–von Mises methods, to estimate the TIHLIW parameters. The performance of these proposed estimation methods is conducted via some simulations. A real data application proves that the TIHLIW model provides consistently better fits compared to some other rival models.



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*Source: 3297693-2020-10-20.xml*