# On Properties of Length-Biased Exponential Model

Authors: Anis Ben Ghorbal
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The type II half logistic length-biased exponential distribution, which extends the length-biased exponential model, is introduced and examined. This model's statistical features, such as moments, conditional moments, mean order statistics, and entropy, are derived. To estimate distribution parameters, statistical inference employs three estimation methods: maximum likelihood, Cramer–von Mises, and Anderson–Darling. To evaluate the performance of the aforementioned estimators, we run simulation experiments based on the graphical results. The TIIHLLBE model's adaptability has been demonstrated through applications to real-world datasets.


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## Body

## 1. Introduction

In the last years, many different statisticians have attracted the attention of generated families of distributions such as Kumaraswamy-G by Cordeiro and De Castro [1], sine-G by Kumar et al. [2], type II half logistic-G [3], exponentiated M-G by Bantan etb al. [4], and Topp-Leone odd Fréchet by Al-Marzouki et al. [5], among others.A new family of continuous distributions with a different scale parameterθ > 0 called the type II half logistic-G (TIIHL-G) family was studied by Hassan et al. [3]. The cumulative distribution function (cdf) of TIIHL-G is expressed by(1)Fx;θ,∅=1−∫0−logGx;∅2λe−θt1+e−θt2dt=2Gx;∅θ1+Gx;∅θ,x>0,θ>0,where Gx;∅ is cdf of baseline model with parameter vector ∅ and Fx;θ,∅ is cdf derived by the T-X generator proposed by Dara and Ahmad [6]. The pdf of the TIIHL-G family is given as(2)fx;θ,∅=2θgx;∅Gx;∅θ−11+Gx;∅θ2,x>0,θ>0,respectively. A random variable X has pdf (2) which will be defined as X∼TIIHL−Gθ,ϕ. Dara and Ahmad [6] proposed the length-biased exponential (LBE) model (or called moment exponential (ME) model) through assigning weight to the exponential (E) model by following the idea of Fisher [7]. They demonstrated that the LBE distribution is more flexible than the exponential (E) model. The cdf and pdf are(3)Gx;β=1−1+xβe−xβ,x>0,and(4)gx,β=xβ2e−xβ,x>0,respectively, where β>0 is a scale parameter. In this paper, we suggest an extension of the LBE model; this extension is constructed by using the TIIHL-G family and LBE model, the so-called type II half logistic length-biased exponential (TIIHLLBE) distribution.A nonnegative random variable x has TIIHLLBE distribution with two parametersβ, θ > 0 which is constructed by inserting (3) and (4) in (1) and (2), and we get the cdf and pdf, respectively:(5)Fx=21−1+x/βe−x/βθ1+1−1+x/βe−x/βθ,θ,β>0x>0,and(6)fx=2θ/β2xe−x/β1−1+x/βe−x/βθ−11+1−1+x/βe−x/βθ2,θ,β>0x>0.The remainder of this article is outlined as follows. In Section2, the linear representation of TIIHLLBE pdf and cdf is presented. The reliability measures of TIIHLLBE model are discussed in Section 3. Basic properties of distribution inclusive moments, moment generating function, the conditional moment, entropy, and probability-weighted moments are calculated in Section 4. In Section 5, its order statistics are derived. Parameter estimation by maximum likelihood (ML), Cramer–von Mises (CVM), and Anderson–Darling (AD) estimation methods is discussed in Section 6. Monte Carlo simulation schemes are performed in Section 7. In Section 8, we applied real datasets to investigate the potentiality of the TIIHLLBE by using some measures of goodness of fit such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (CAIC), Hannan–Quinn Information Criterion (HQIC), and Kolmogorov–Smirnov (KS) statistic (with its p value (PV). Finally, conclusion is given in Section 9.


## 2. Linear Representation

In this section, a linear representation of the pdf and cdf is presented to introduce statistical properties of TIIHLLBE distribution. Use the next binomial expansion(7)1+z−b=∑i=0∞−1ib+i−1izi,where |z|<1 and b is a positive real noninteger. By applying (7) in the next term, we get(8)1+1−1+xβe−x/βθ−2=∑i=0∞−1ii+11−1+xβe−x/βθi.Inserting the previous equation in (6), we have(9)fx=2θβ2∑i=0∞−1ii+1xe−xβ1−1+xβe−xβθi+1−1.Again applying the general binomial theorem, we get(10)1−1+xβe−xβθi+1−1=∑i=0∞−1jθi+1−1j1+xβje−jxβ.Again using the binomial expansion, we get(11)fx=∑k=0∞w¯kxk+1e−j+1xβ,where w¯k=2θ/β2+k∑i,j=0∞−1i+ji+1jkθi+1−1j.A linear representation for the cdf can be calculated by using binomial expansion forFxs, where s is an integer:(12)Fxs=2s1−1+x/βe−x/βθs1+1−1+x/βe−x/βθs.By applying (7) in the next term, we get(13)1+1−1+xβe−xβθ−s=∑m=0s−1ms+m−1m1−1+xβe−xβθm.Then,(14)Fxs=2s∑m=0s−1ms+m−1m1−1+xβe−xβθm+s.Again applying the binomial expansion to the next term,(15)1−1+xβe−xβθm+s=∑l=0∞−1lθs+ml1+xβle−lxβ.Then,(16)Fxs=2s∑m,l=0s−1m+ls+m−1mθs+ml1+xβle−lxβ.Again inserting the binomial expansion, we get(17)Fxs=∑h=0∞Ψhxhe−lxβ,where Ψh=2s∑m,l=0s−1m+lβ−hs+m−1mθs+mllh.


## 3. The Reliability Measures of Length-Biased Exponential Distribution

In this section, we introduce the reliability measures such as survival function, hazard rate function, reversed hazard rate function, residual, and reversed residual lifetime. The reliability function (survival function) of TIIHLLBE distribution is given by(18)F¯x=1−1−1+x/βe−x/βθ1+1−1+x/βe−x/βθ.The hazard rate (hrf) function or failure rate and reversed hrf for the TIIHLLBE are obtained by(19)hx=2θ/β2xe−x/β1−1+x/βe−x/βθ−11−1−1+x/βe−x/β2θ,and(20)τx=θ/β2xe−x/β1−1+x/βe−x/β1+1−1+x/βe−x/βθ.The plots of the pdf, cdf, and survival and hazard rate functions of TIIHLLBE with different values of parameters are mentioned in Figures1–4.Figure 1
Plots of pdf for TIIHLLBE model.Figure 2
Plots of cdf for TIIHLLBE model.Figure 3
Plots of survival for TIIHLLBE model.Figure 4
Plots of hrf for TIIHLLBE model.Thenth moment of the residual life, say znt, is given by(21)znt=Ex−tn=∫t∞x−tndFx1−Ft,x>t,n=1,2,….Then,(22)znt=1F¯t∑t=0∞∑r=0n−1rn!r!n+r!w¯ktrβj+1n+k−r+2Γn+k−r+2,j+1tβ,where Γs,t=∫t∞xs−1e−xdx is the upper incomplete gamma function.Now, thenth moment of the reversed residual life is calculated as(23)Znt=Et−xn=1Ft∫0tt−zndFz,z≤t,t>0,n=1,2,….Then,(24)znt=1Ft∑k=0∞∑r=0nnr−1rw¯ktn−rβj+1r+k+2γr+k+2,j+1tβ,where γs,t=∫0txs−1e−xdx is the lower incomplete gamma function.


## 4. Properties

In this section, we introduce some properties of TIIHLLBE distribution such as moments, moment generating function, the conditional moment, entropy, and probability-weighted moments (PMWs).


### 4.1. Moments

Therth moment of x denoted by μ`r can be derived from (6) as follows:(25)μ`r=∑k=0∞w¯k∫0∞xr+k+1e−j+1xβdx.Then,(26)μ`r=∑k=0∞w¯kβj+1r+k+2Γr+k+2.Whenr = 1, we get EX=∑k=0∞w¯kβ/j+1k+3Γk+3.The moment generating function ofxMxt is given from (6) as(27)Mxt=Eetx=∑k=0jw¯k∫0∞xk+1e−xj+1/β−tdx=Γk+2j+1/β−tk+2.Numerical values for specific values of parameters of the first four ordinary moments:μ1′,μ2′,μ3′,μ4′, variance σ2, skewness (SK), and kurtosis (KU) of the TIIHLLBE model are listed in Tables 1 and 2.Table 1
Numerical values of moments for the TIIHLLBE model atβ=0.5.
μr'θ=0.50θ=0.10θ = 1.5θ = 2.0θ = 2.5θ = 3.0θ = 3.5μ1'1.4321.3441.2411.1180.9640.7620.469μ2'2.4982.251.9781.6771.3390.9550.509μ3'5.284.6393.9673.2582.5091.7150.875μ4'13.38111.6039.7817.9135.9964.0342.03σ20.4480.4440.4380.4280.410.3740.289SK1.4031.431.4691.5281.6271.822.35KU6.2976.3886.5246.7447.1398.00710.966CV0.4670.4960.5330.5850.6640.8031.145Table 2
Numerical values of moments for the TIIHLLBE model atβ=2.
μr'β = 0.5β = 1.0β = 1.5β = 2.0β = 2.5β = 3.0β = 3.5μ1'7.8246.7065.5884.4713.3532.2351.118μ2'82.17260.37141.92426.83215.0936.7081.677μ3'1118703.755407.266208.5287.96926.0653.258μ4'19001025049452026640.913126.67.913σ220.96315.40110.6956.8453.851.7110.428SK1.5281.5281.5281.5281.5281.5281.528KU6.7446.7446.7446.7446.7446.7446.744CV0.5850.5850.5850.5850.5850.5850.585


### 4.2. Conditional Moment

LetX denote a random variable with the pdf given in (6). The sth upper incomplete moment, say ηst, can be derived as(28)ηst=∫t∞xsfxdx=∑k=0∞w¯k∫t∞xs+k+1e−j+1xβdx=∑k=0∞w¯kβj+1s+k+2Γs+k+2,j+1tβ,where Γs,t=∫t∞xs−1e−xdx is the upper incomplete gamma function.Similarly, thesth lower incomplete moment function is given by(29)ϕst=∫t∞xsfxdx=∑k=0∞w¯k∫t∞xs+k+1e−j+1xβdx=∑k=0∞w¯kβj+1s+k+2γs+k+2,j+1tβ,where γs,t=∫0txs−1e−xdx is the lower incomplete gamma function.


### 4.3. Entropy

Ifx has the pdf given in (6), then the Rényi entropy is defined by(30)IRρ=11−ρLog∫0∞fρxdx,ρ≠1,ρ>0.By using (6), then(31)fx=2θβ2ρxρe−ρxβ1−1+xβe−xβρθ−11+1−1+xβe−xβθ−2ρ.By using the binomial expansion more times for the last equation, we get(32)fρx=∑k=0∞ηkxρ+ke−ρ+jxβ,where ηk=2θρ/β2ρ+k∑i,j=0∞jk−1i+jΓ2ρ+iΓθρ+i−ρ+1/i!j!Γ2ρΓθρ+i−j−ρ+1. Then,(33)∫0∞fρxdx=∑k=0∞ηk∫0∞xρ+ke−ρ+jxβdx=∑k=0∞ηkβρ+jρ+k+1Γρ+k+1.The Rényi entropy of TIIHLLBE distribution is(34)IRρ=11−ρLog∑k=0∞ηkβρ+jρ+k+1Γρ+k+1.


### 4.4. Probability-Weighted Moments

The PWMsMs of X, say τr,s, are defined by the formula(35)τr,s=ExrfxFsx=∫−∞∞xrfxFsxdx.The PWMsMs of TIIHLLBE distribution are obtained by substituting (11) and (17) into (35), and Fsx is defined as follows:(36)τr,s=∑k,h=0∞w¯kΨh∫0∞xr+h+k+1e−l+j+1xβdx.Then,(37)τr,s=∑k,h=0∞w¯kΨhβl+j+1r+h+k+2Γr+h+k+2.


## 4.1. Moments

Therth moment of x denoted by μ`r can be derived from (6) as follows:(25)μ`r=∑k=0∞w¯k∫0∞xr+k+1e−j+1xβdx.Then,(26)μ`r=∑k=0∞w¯kβj+1r+k+2Γr+k+2.Whenr = 1, we get EX=∑k=0∞w¯kβ/j+1k+3Γk+3.The moment generating function ofxMxt is given from (6) as(27)Mxt=Eetx=∑k=0jw¯k∫0∞xk+1e−xj+1/β−tdx=Γk+2j+1/β−tk+2.Numerical values for specific values of parameters of the first four ordinary moments:μ1′,μ2′,μ3′,μ4′, variance σ2, skewness (SK), and kurtosis (KU) of the TIIHLLBE model are listed in Tables 1 and 2.Table 1
Numerical values of moments for the TIIHLLBE model atβ=0.5.
μr'θ=0.50θ=0.10θ = 1.5θ = 2.0θ = 2.5θ = 3.0θ = 3.5μ1'1.4321.3441.2411.1180.9640.7620.469μ2'2.4982.251.9781.6771.3390.9550.509μ3'5.284.6393.9673.2582.5091.7150.875μ4'13.38111.6039.7817.9135.9964.0342.03σ20.4480.4440.4380.4280.410.3740.289SK1.4031.431.4691.5281.6271.822.35KU6.2976.3886.5246.7447.1398.00710.966CV0.4670.4960.5330.5850.6640.8031.145Table 2
Numerical values of moments for the TIIHLLBE model atβ=2.
μr'β = 0.5β = 1.0β = 1.5β = 2.0β = 2.5β = 3.0β = 3.5μ1'7.8246.7065.5884.4713.3532.2351.118μ2'82.17260.37141.92426.83215.0936.7081.677μ3'1118703.755407.266208.5287.96926.0653.258μ4'19001025049452026640.913126.67.913σ220.96315.40110.6956.8453.851.7110.428SK1.5281.5281.5281.5281.5281.5281.528KU6.7446.7446.7446.7446.7446.7446.744CV0.5850.5850.5850.5850.5850.5850.585


## 4.2. Conditional Moment

LetX denote a random variable with the pdf given in (6). The sth upper incomplete moment, say ηst, can be derived as(28)ηst=∫t∞xsfxdx=∑k=0∞w¯k∫t∞xs+k+1e−j+1xβdx=∑k=0∞w¯kβj+1s+k+2Γs+k+2,j+1tβ,where Γs,t=∫t∞xs−1e−xdx is the upper incomplete gamma function.Similarly, thesth lower incomplete moment function is given by(29)ϕst=∫t∞xsfxdx=∑k=0∞w¯k∫t∞xs+k+1e−j+1xβdx=∑k=0∞w¯kβj+1s+k+2γs+k+2,j+1tβ,where γs,t=∫0txs−1e−xdx is the lower incomplete gamma function.


## 4.3. Entropy

Ifx has the pdf given in (6), then the Rényi entropy is defined by(30)IRρ=11−ρLog∫0∞fρxdx,ρ≠1,ρ>0.By using (6), then(31)fx=2θβ2ρxρe−ρxβ1−1+xβe−xβρθ−11+1−1+xβe−xβθ−2ρ.By using the binomial expansion more times for the last equation, we get(32)fρx=∑k=0∞ηkxρ+ke−ρ+jxβ,where ηk=2θρ/β2ρ+k∑i,j=0∞jk−1i+jΓ2ρ+iΓθρ+i−ρ+1/i!j!Γ2ρΓθρ+i−j−ρ+1. Then,(33)∫0∞fρxdx=∑k=0∞ηk∫0∞xρ+ke−ρ+jxβdx=∑k=0∞ηkβρ+jρ+k+1Γρ+k+1.The Rényi entropy of TIIHLLBE distribution is(34)IRρ=11−ρLog∑k=0∞ηkβρ+jρ+k+1Γρ+k+1.


## 4.4. Probability-Weighted Moments

The PWMsMs of X, say τr,s, are defined by the formula(35)τr,s=ExrfxFsx=∫−∞∞xrfxFsxdx.The PWMsMs of TIIHLLBE distribution are obtained by substituting (11) and (17) into (35), and Fsx is defined as follows:(36)τr,s=∑k,h=0∞w¯kΨh∫0∞xr+h+k+1e−l+j+1xβdx.Then,(37)τr,s=∑k,h=0∞w¯kΨhβl+j+1r+h+k+2Γr+h+k+2.


## 5. Order Statistics

Order statistics have been used in many applications of statistics, such as life testing and reliability. LetX1,X2,…,Xn be a random sample of size n from the pdf given by (6) and the cdf given by (5). Let X1:n<X2:n<⋯<Xn:n be the ordered random sample from a TIIHLLBE of size n. Then, the pdf of Xr:n,1≤r≤n, is given by(38)fr:nx=n!r−1!n−r!∑v=0∞fxFxv+r−1,−∞<x<∞.By inserting (11) and (17) in (38) and replacing s by v+r−1, we get(39)fr:nx=∑k,h,v=0∞Cr:nxh+k+1e−l+j+1xβ,−∞<x<∞,where Cr:n=−1vn−rvn!/r−1!n−r!w¯kΨh.Now, we have derived an exact expression for the single moments of order statistics from TIIHLLBE. We shall first establish the exact expression forEXr:np=μr:np. Using (39), we obtain the pth moment of rth order statistics as follows:(40)EXr:np=∑k,h,v=0∞Cr:n∫0∞xp+h+k+1e−l+j+1xβdx,−∞<x<∞.Then, we get(41)EXr:np=∑k,h,v=0∞Cr:nβl+j+1p+h+k+2Γp+h+k+2.Forp=1, in (41), we get mean of rth order statistics.(42)EXr:n=∑k,h,v=0∞Cr:nβl+j+1h+k+3Γh+k+3.The jointpdf of Xr:n and Xs:n, 1≤r≤s≤n, is given by(43)fr,s:nx,y=Cr,s:nFxr−1Fy−Fxs−r−11−Fyn−sfxfy,−∞<x<y<∞,where Cr,s:n=n!/r−1!s−r−1!n−s!,xr:n=x,andxs:n=y.The exact expression forEXr:npXs:nq=μr,s:np,q. Using (43), we obtain the pth and qth moments of order statistics as follows:(44)μr,s:np,q=Cr,s:n∫0∞∫x∞xpyqFxr−1Fy−Fxs−r−11−Fyn−sfxfydxdy.By using the binomial expansion, we get(45)μr,s:np,q=Cr,s:n∑i=0s−r−1∑j=0n−s−1i+js−r−1in−sj∫0∞xpFxr+i−1Ixfxdx,where Ix=∫x∞yqFys−r−i+j−1fydy.By using (11) and (17) and replacing s by s−r−i+j−1, we get(46)Ix=2s−r−i+jθβp∑k=0s−r−i+j+1∑l=0θs−r−i+k+l∑m=0l−1ls−r−i+jk,θs−r−i+j+k+lllm1l+1p+m+2Γm+p+2,xl+1β.Substituting the value ofIx in (45) and simplifying the resulting equation, we get(47)μr,s:np,q=Cr,s:n∑i=0s−r−1∑j=0n−s∑k=0s−r−i+j+1∑l=0θs−r−i+k+l∑m=0l∑n=0m+p+1∑u=0r+i+1∑ν=0θr+u+1−1∑ω=0ν−1i+j+l+u,2s−r−i+jθβps−r−1in−sjs−r−i+jkθs−r−i+j+k+ll,1l+1p+m+2m+p+1!n!l+1βnr+i−1uθr+u+i−1ννω,1βωΓp+n+ω+2l+ν+2/βp+n+ω+2.


## 6. Statistical Inference

In this section, we propose several approaches to estimate the parametersθ and β of TIIHLLBE distribution model using maximum likelihood (ML), Cramer–von Mises (CVM), and Anderson–Darling (AD) methods of estimation.


### 6.1. Method of Maximum Likelihood

Letx1,x2,…,xn be a random sample of size n from TIIHLLBE model with parameters θ and β, and the log-likelihood function is(48)LX|β,θ=nln2θ−2nlnβ+∑i=1nxi−∑i=1nxiβ+θ−1∑i=1nlog1−1+xiβe−xiβ−2∑i=1nlog1+1−1+xiβ1−e−xiβθ.For calculation of MLE estimation, we need partial derivatives ofLX|β,θ by parameters(49)∂logLX|β,θ∂θ=nθ+∑i=1nlogZi−2∑i=1nZiθlnZi1+Ziθ,and(50)∂logLX|β,θ∂θ=−2nβ+∑i=1nxiβ2+θ−1∑i=1nxi/β22+xi/βe−xi/βZi−2∑i=1nθZiθ−1xi/β22+xi/βexi/β1+Ziθ,where Zi=1−1+xi/βe−xi/β. The estimations of the parameters can be found by par^MLE as solution of the system(51)∂LX|β,θ∂β=0;∂LX|β,θ∂θ=0..


### 6.2. Method of Minimum Distance

In this part, we consider well-known CVM-type minimum distance estimators [8–10], which are based on the next statistic:(52)CVMβ,θ=∑i=1nFxi:n;β,θ−2i−12n2.Using form of the objective function, we can find that(53)∂CVMβ,θ∂β=−2θβ3∑i=1n2∇iθ1+∇iθ−2i−12n∇iθ−1xi:n2exi:n/β1+∇iθ2,and(54)∂CVMβ,θ∂θ=2θ∑i=1n2∇iθ1+∇iθ−2i−12n∇iθ−11+∇iθ2,where ∇i=1−1+xi:n/βe−xi:n/β. The estimations of the parameters can be found by par^CVM as solution of the system(55)∂CVMβ,θ∂β=0;∂CVMβ,θ∂θ=0..


### 6.3. Anderson–Darling Method

The method of Anderson–Darling estimation was studied in [11] in the context of statistical tests.(56)ADβ,θ=−n−1n∑i=1n2i−1logFxi:n|β,θ+logF¯xn+1−i:n|β,θ,where F¯xn+1−i:n|β,θ=1−Fxn+1−i:n|β,θ.For minimizing the function,ADβ,θ, we need to find partial derivatives by parameters β,θ:(57)∂ADβ,θ∂β=−2θnβ3∑i=1n2i−1xi:n2e−xi:n/β2∇i1+∇iθ−Δiθ−1xn+1−i:n2e−xn+1−i:n/β1−Δi2θ,and(58)∂ADβ,θ∂β=−2θn∑i=1n2i−112∇i1+∇iθ2−Δiθ−11−Δi2θ,where Δi=1−1+xn+1−i:n/βe−xn+1−i:n/β. In results estimations of the parameters can be find par^AD as solution of the system(59)∂ADβ,θ∂β=0;∂ADβ,θ∂θ=0..


## 6.1. Method of Maximum Likelihood

Letx1,x2,…,xn be a random sample of size n from TIIHLLBE model with parameters θ and β, and the log-likelihood function is(48)LX|β,θ=nln2θ−2nlnβ+∑i=1nxi−∑i=1nxiβ+θ−1∑i=1nlog1−1+xiβe−xiβ−2∑i=1nlog1+1−1+xiβ1−e−xiβθ.For calculation of MLE estimation, we need partial derivatives ofLX|β,θ by parameters(49)∂logLX|β,θ∂θ=nθ+∑i=1nlogZi−2∑i=1nZiθlnZi1+Ziθ,and(50)∂logLX|β,θ∂θ=−2nβ+∑i=1nxiβ2+θ−1∑i=1nxi/β22+xi/βe−xi/βZi−2∑i=1nθZiθ−1xi/β22+xi/βexi/β1+Ziθ,where Zi=1−1+xi/βe−xi/β. The estimations of the parameters can be found by par^MLE as solution of the system(51)∂LX|β,θ∂β=0;∂LX|β,θ∂θ=0..


## 6.2. Method of Minimum Distance

In this part, we consider well-known CVM-type minimum distance estimators [8–10], which are based on the next statistic:(52)CVMβ,θ=∑i=1nFxi:n;β,θ−2i−12n2.Using form of the objective function, we can find that(53)∂CVMβ,θ∂β=−2θβ3∑i=1n2∇iθ1+∇iθ−2i−12n∇iθ−1xi:n2exi:n/β1+∇iθ2,and(54)∂CVMβ,θ∂θ=2θ∑i=1n2∇iθ1+∇iθ−2i−12n∇iθ−11+∇iθ2,where ∇i=1−1+xi:n/βe−xi:n/β. The estimations of the parameters can be found by par^CVM as solution of the system(55)∂CVMβ,θ∂β=0;∂CVMβ,θ∂θ=0..


## 6.3. Anderson–Darling Method

The method of Anderson–Darling estimation was studied in [11] in the context of statistical tests.(56)ADβ,θ=−n−1n∑i=1n2i−1logFxi:n|β,θ+logF¯xn+1−i:n|β,θ,where F¯xn+1−i:n|β,θ=1−Fxn+1−i:n|β,θ.For minimizing the function,ADβ,θ, we need to find partial derivatives by parameters β,θ:(57)∂ADβ,θ∂β=−2θnβ3∑i=1n2i−1xi:n2e−xi:n/β2∇i1+∇iθ−Δiθ−1xn+1−i:n2e−xn+1−i:n/β1−Δi2θ,and(58)∂ADβ,θ∂β=−2θn∑i=1n2i−112∇i1+∇iθ2−Δiθ−11−Δi2θ,where Δi=1−1+xn+1−i:n/βe−xn+1−i:n/β. In results estimations of the parameters can be find par^AD as solution of the system(59)∂ADβ,θ∂β=0;∂ADβ,θ∂θ=0..


## 7. Monte Carlo Simulation

A Monte Carlo simulation study is studied to evaluate the behavior of the estimates with respect to their bias and root mean square error (RMSE). We calculate next characteristic of the estimations:(60)Biasβ^,θ^=1N∑i=1Nβ^i,θ^i−β,θ,RMSEβ^=1N∑i=1Nβ^i−β2,RMSEθ^=1N∑i=1Nθ^i−θ2.Consider results of the Monte Carlo simulation for parametersβ=0.5,θ=1.5 with generated random sample N=10000 simulations from TIIHLLBE distribution and sample size n=25,100,200,500.The bias and RMSE of the estimates of the unknown parameters are evaluated. Monte Carlo simulation results are mentioned in Table3.Table 3
Bias and RMSE of TIIHLLBE distribution for ML, CV, and AD estimates.
nMethodBiasβ^RMSEβ^Biasθ^RMSEθ^25MLE−0.102150.79538−0.32040.585925CM−0.016580.132580.3502050.85268AD0.0026270.118460.1614440.567845100MLE−0.053780.12112−0.0230.402CM−0.0030.06520.056270.323AD0.003080.059840.0457010.26625200MLE−0.002880.042830.045440.18325CM−0.003240.052210.044350.22357AD0.000560.047390.028920.20192500MLE−0.00160.03850.040.16342CM−0.00180.04470.034470.1422AD0.00020.026740.01650.17435


## 8. Applications

In this section, we compare the TIIHLLBE with some other known competitive models to demonstrate its importance in data modeling. The MLE method is used to estimate the parameters of the competitive models. The AIC, BIC, CAIC, HQIC, KS, and PV model selection criteria and goodness-of-fit tests are used to choose best model.Data I.
The first dataset represents the survival times of 72 guinea pigs (in days) infected with virulent tubercle bacilli, and these data were introduced by Bjerkedal [12].Data II.
The second dataset (called hazard time data) consists of 20 patients referring to the lifetime and relating to relief times (in minutes). Patients who received an analgesic were introduced by Gross and Clark in [13].
Some descriptive statistics of both datasets are mentioned in Table4, and the boxplots are shown in Figure 5.
The total time test (TTT) plot (see [14]) is an important plot approach to verify whether the data can be applied to a specific model or not. The TTT plots of two real datasets are shown in Figure 6, and this plot shows that the empirical hrf of both datasets is increasing, as shown in Figure 7.
We will compare the fits of the TIIHLLBE model with those of other competitive models, namely: Marshall-Olkin E (MOE), Burr X-E (BrXE), Kumaraswamy E (KuE), beta E (BE), Kumaraswamy Marshall-Olkin E (KuMOE), generalized Marshall-Olkin E (GMOE), Marshall-Olkin Kumaraswamy E (MOKuE), ME, and E models (see [15]).
The MLEs and standard errors (SEs) are calculated via these two datasets, and numerical results are mentioned in Tables4 and 5. Depending on the numerical values which is obtained in Tables 6 and 7 along with Figures 8 and 9, the TIIHLLBE model is much better than the above-mentioned extensions of the E model, so the TIIHLLBE model is a good alternative to these models in both datasets (Table 8).
The parameters of TIIHLLBE distribution are estimated by using three different estimation methods, ML, CVM, and AD estimation methods. The efficiency of the estimation methods is the same in the two datasets as shown in Tables9 and 10. We see that the CVM estimation method is the best among the others in the two datasets.Table 4
Descriptive statistics for both datasets.
nMeanMedianModeVarSKKUData I721.7681.4951.081.0551.3712.225Data II201.91.71.70.4711.8624.185Figure 5
Boxplots for both datasets.Figure 6
TTT plots for both datasets.Figure 7
Estimated hrfs for both datasets.Table 5
Some measures of goodness of fit for data I.
ModelsAIC,BIC,CAIC,HQIC,KS(PV)TIIHLLBE192.209191.924192.383194.0220.089(0.619)KuMOE207.82216.94208.42211.420.09(0.53)BrXE235.30239.90235.50237.100.22(0.002)GMOE210.54217.38210.89213.240.09(0.51)BE207.38214.22207.73210.080.11(0.34)KuE209.42216.24209.77212.120.09(0.50)MOKuE209.44218.56210.04213.040.10(0.44)MOE210.36214.92210.53212.160.10(0.43)ME210.40212.68210.45211.300.14(0.13)E234.63236.91234.68235.540.27(0.06)Table 6
Estimates for data I.
ModelsMLEs and SEsTIIHLLBE (θ, β)1.845, 0.83 (0.317), (0.1005)KuMOE (α, a, b, β)0.3731, 3.4782, 3.3 063, 0.2990 (0.1358), (0.862), (0.781), (1.113)BrXE (θ, β)0.475, 0.2055 (0.060), (0.012)GMOE (λ, α, β)0.1789, 47.6350, 4.4652 (0.0702), (44.9011), (1.3270)BE (a, b,β)0.8073, 3.4612, 1.3311 (0.6961), (1.0032), (0.8551)KuE (a, b,β)3.3041, 1.1002, 1.0371 (1.1061), (0.7642), (0.6141)MOKuE (α, a, b, β)0.0081, 2.7162, 1.9861, 0.0992 (0.0021), (1.3158), (0.7839), (0.0481)MOE (α, β)8.7776, 1.3788 (3.5549), (0.1929)ME (β)0.9252 (0.0768)E (β)0.540 (0.063)Table 7
Estimates for data II.
ModelsMLEs and SEsTIIHLLBE (θ, β)12.326, 0.432 (6.897), (0.0812)KuMOE (α, a, b, β)8.8679, 34.8258, 0.2989, 4.8988 (9.1459), (22.3119), (0.2387), (3.1757)BrXE (θ, β)1.1635, 0.3207 (0.330), (0.030)GMOE (λ, α, β)0.5192, 89.4623, 3.1691 (0.2561), (66.2782), (0.7721)BE (a, b,β)81.6333, 0.5421, 3.5142 (120.4104), (0.3272), (1.4101)KuwE (a, b,β)83.7558, 0.5679, 3.3329 (42.3612), (0.3261), (1.1880)MOKuE (α, a, b, β)0.1333, 33.2322, 0.5711, 1.6691 (0.3320), (57.8371), (0.7211), (1.8141)MOE (α, β)54.4739, 2.3159 (35.5817), (0.3738)ME (β)0.9502 (0.1501)E (β)0.526 (0.117)Figure 8
Estimated cdf, pdf, pp plots and estimated for data I.Figure 9
Estimated cdf, pdf, pp plots and estimated for data II.Table 8
Some measures of goodness of fit for data II.
ModelsAIC,BIC,CAIC,HQIC,KS(PV)TIIHLLBE35.99634.59936.70236.3850.123(0.923)MOE43.5145.5144.2243.900.18(0.55)BrXE48.1050.1048.8048.500.25(0.17)KuE41.7844.7543.2842.320.14(0.86)GMOE42.7545.7444.2543.340.15(0.78)BE43.4846.4544.9844.020.16(0.80)KuMOE42.8046.8445.5543.600.15(0.86)MOKuE41.5845.5444.2542.300.14(0.87)ME54.3255.3154.5454.500.32(0.07)E67.6768.6767.8967.870.44(0.004)Table 9
The goodness-of-fit statistics for parameters estimated under various methods for the first dataset.
Methodθ^β^KSPVMLE1.8450.830.0890.619CVME2.2070.7440.0690.883ADE2.0860.7730.0730.838


## 9. Concluding Remarks

In this article, a new model, called TIIHLLBE distribution, is introduced. Some statistical properties of the proposed distribution such as moments, order statistics, conditional moments, probability-weighted moment, and Rényi entropy are derived and discussed. The estimation of the model parameters is discussed through ML, CVM, and AD methods. Monte Carlo simulation study is carried out to compare the performance of the three different estimates. Applications to two real datasets indicate that the new model is superior to the fits when compared to the other well-known distributions.Table 10
The goodness-of-fit statistics for parameters estimated under various methods for the second dataset.
Methodθ^β^KSPVMLE12.3260.4320.1230.923CVME15.5980.3950.0880.998ADE11.9250.4310.1070.976



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