# Bayesian Analysis of Record Statistic from the Inverse Weibull Distribution under Balanced Loss Function

Authors: Fuad S. Al-Duais
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The main contribution of this work is to develop a linear exponential loss function (LINEX) to estimate the scale parameter and reliability function of the inverse Weibull distribution (IWD) based on lower record values. We do this by merging a weight into LINEX to produce a new loss function called weighted linear exponential loss function (WLINEX). We then use WLINEX to derive the scale parameter and reliability function of the IWD. Subsequently, we discuss the balanced loss functions for three different types of loss function, which include squared error (SE), LINEX, and WLINEX. The majority of previous scholars determined the weighted balanced coefficients without mathematical justification. One of the main contributions of this work is to utilize nonlinear programming to obtain the optimal values of the weighted coefficients for balanced squared error (BSE), balanced linear exponential (BLINEX), and balanced weighted linear exponential (BWLINEX) loss functions. Furthermore, to examine the performance of the proposed methods—WLINEX and BWLINEX—we conduct a Monte Carlo simulation. The comparison is between the proposed methods and other methods including maximum likelihood estimation, SE loss function, LINEX, BSE, and BLINEX. The results of simulation show that the proposed models BWLINEX and WLINEX in this work have the best performance in estimating scale parameter and reliability, respectively, according to the smallest values of mean SE. This result means that the proposed approach is promising and can be applied in a real environment.


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## Body

## 1. Introduction

Statistics related to record values are of interest for many real-life applications. For instance, they allow prediction of the possible time of earthquakes, floods, extreme weather events, and life-testing studies. Many scholars have researched record values and associated statistics, including generalized extreme value distribution [1], Lomax distribution [2], Weibull distribution [3]; Yang et al [4, 5], log-normal distribution [6], ratio of Weibull random variables [7], power Lindley model [8], exponential distribution [9], generalized Rayleigh model [10], and inverse Weibull distribution (IWD) [11–16]. In this work, we will investigate IWD based on record values.IWD is one of the most widely used probability distributions with many real environment applications. This refers to the ability of IWD to model a variety of failure characteristics, such as wear-out periods, useful life, infant mortality, and engineering discipline. The probability density function (PDF) and cumulative distribution function (CDF) of IWD are given as follows, respectively:(1)fx=λθexp−λx−θ,x≥0λ,θ>0,(2)Fx=exp−λx−θ,x≥0,λ,θ>0.The reliability function is given as follows:(3)Rt=1−exp−λt−θ,t≥0,λ,θ>0.Here,λ and θ are scale and shape parameters, respectively.To estimate the parameters and reliability of IWD, scholars use many approaches including Bayesian and non-Bayesian. Many researchers attempt to estimate parameters and reliability depending on squared error (SE) loss function. The main criticism of this approach is that SE gives overestimation and underestimation equal importance. Thus, an alternative loss function is needed. One of these alternatives is the linear exponential (LINEX) loss function, which many authors have discussed, including Calabria and Pulcini [17], Gencer and Saraçoğlu [18], Khatun and Matin [19], and Parsian and Kirmani [20].A new type of loss function called balanced loss function appeared with the aim to utilize the positive criteria of two methods. Balanced loss functions habitually consist of the sum of two estimation methods with different weights. These include balanced squared error (BSE) loss function and balanced linear exponential (BLINEX) loss function [21–23].


## 2. Methodology

The weighted coefficients (ω1 and ω2) in balanced loss functions are routinely determined by arbitrary choice without depending upon any mathematical justification. This motivated us to propose a justifiable mathematical approach to determine these coefficients. The proposed approach is the nonlinear programming by minimizing the mean square error (MSE) function with conditions related to weighted coefficients.Furthermore, we developed a new loss function, which we named weighted linear exponential (WLINEX), by weighting LINEX. We then derived scale parameter and reliability function of the IWD depending on WLINEX. In addition, we employed WLINEX to produce the balanced weighted linear exponential loss function (BWLINEX).


### 2.1. Record Values and Maximum Likelihood Estimation

LetX1,X2,X3,…… be a sequence of independent and identically distributed random variables with CDF Fx and PDF fx. Set Yn=minX1,X2,X3,…,Xn, n≥1, and say that Xj is a lower record and denoted by XLi if Yj<Yj−1,j>1. Suppose we observe the first n lower record values XL1,XL2,XL3,…,XLn from the IWD whose PDF and CDF are given by (1) and (2), respectively. Based on those lower record values, we have the joint density function of the first n lower record values X≡xL1,xL2,xL3,……,xLn as given by Sultan [24]:(4)f1,2,3,……,nxL1,xL2,xL3,……,xLn=fxLn∏i=1n−1fxLi1−FxLi,0≤xL1<xL2<xL3,……<xLn<∞.Here,f. and F. are given by (1) and (2), respectively, after replacing x by xLi. The likelihood function based on the n lower record values x is given as follows:(5)ℓλ,θ|X=λθnuexp−λTm,u=∏i=1nxLi−θ−1,Tm=xLn−θ.We obtain that the log-likelihood function may be written as follows:(6)Lλ,θ|X=lnℓ=nlnλ+nlnθ−λTm−θ+1∑i=1nlnxLi.Assuming that the shape parameterθ is known, using equation (6), the maximum likelihood estimator (MLE) λ^ML of the scale parameter λ can be shown to be of the following form:(7)λ^ML=nxLn−θ.Ifλ is replaced by λ^ML in equation (3), we can obtain the MLE of reliability function R^MLt of R (t) depending on the invariance property:(8)R^MLt=1−exp−λ^MLt−θ.


### 2.2. Loss Functions

In the following sections, we present the four main types of loss functions under investigation in this work.


#### 2.2.1. Squared Error Loss Function

The SE loss function can be expressed as(9)Lφ^,φ=φ^−φ2.The Bayes estimator ofφ based on SE loss function can be obtained as follows:(10)φ^SE=Eπφ|X.


#### 2.2.2. Linear Exponential Loss Function

Varian [25] introduced the LINEX loss function. LINEX is an asymmetric loss function that can be expressed as(11)LΔ∝expcΔ−cΔ−1,c≠0,where Δ=φ^−φ. The sign and magnitude of c reflect the direction and degree of asymmetry, respectively. The Bayes estimator relative to LINEX loss function, denoted by φ^LINEX, is given as follows:(12)φ^LINEX=−1cLnEφexp−cφ,c≠0,provided that Eφ=e−cφ exists and is finite, where Eφ denotes the expected value.


#### 2.2.3. Weighted Linear Exponential Loss Functions

The researcher proposes this loss function depending on WLINEX loss function as follows:(13)L⋆Δ=L⋆φ^−φ∝wφexpcΔ−cΔ−1,c≠0.Here,φ^ represents the estimated parameter that makes the expectation of loss function by equation (13) as small as possible. The value wφ represents the proposed weighted function, which is equal to the following:(14)wφ=exp−zφ.Depending on the posterior distribution of the parameterφ and using the proposed weighted function as in equation (14), we can attain the estimated weighted Bayes of the parameter φ as follows:(15)ELw⋆φ^,φ=∫∀φLwφ^,φfφ|Xdφ=∫∀φwφexpcφ^−φ−cφ^−φ−1fφ|Xdφ=∫∀φexp−zφexpcφ^−φfφ|Xdφ−∫∀φexp−zφcφ^−φfφ|Xdφ−∫∀φexp−zφfφ|Xdφ=expcφ^∫∀φexp−φz+cfφ|Xdφ−cφ^∫∀φexp−zφfφ|Xdφ+c∫∀φφexp−zφfφ|Xdφ−∫∀φexp−zφfφ|Xdφexpcφ^Eφexp−φz+c|X−cφ^Eφexp−zφ|X+cEφφexp−zφ|X−Eφexp−zφ|X.It is known that, to find the value ofφ^ that minimizes ELw⋆φ^,φ, we have to perform the following two steps:(i)∂EφLw⋆φ^,φ/∂φ^=0(16)∂EφLw⋆φ^,φ∂φ^=cexpcφ^Eφexp−φz+c|X−cEφexp−zφ|X=0.Therefore, we can find the following:(17)c=expcφ^Eφexp−φz+c|X=cEφexp−zφ|X.Consequently, the Bayesian estimation of the parameterφ using WLINEX will be(18)φ^WLINEX=1cLnEφexp−zφ|XEφexp−φz+c|X,z+c≠0.(i)
Eφ=e−zφ and Eφ=e−z+cφ exist and are finite, where Eφ denotes the expected value.(ii
)∂2EφLw⋆φ^,φ/∂φ^2>0 at the minimum value computed by (i):(19)∂2EφLw⋆φ^,φ∂φ^2=∂∂φ^∂EφLw⋆φ^,φ∂φ^=c2expcφ^Eφexp−φz+c>0.Becauseφ^WLINEX satisfies conditions (i) and (ii), it follows that φ^WLINEX is the minimum value. Note that the WLINEX loss function is a generalization of the LINEX loss function, where LINEX is a special case of WLINEX when z=0 in equation (18).


#### 2.2.4. Balanced Loss Function

According to AbdEllah [22], the class of balanced loss function (BLF) can be written in the form(20)Lρ,ω,τ0kξφ,τ=ω1kφρτ0,τ+ω2kφρξφ,τ,ω1+ω2=1,where τ represents an estimator of parameter ξφ,τ0 is a chosen prior estimator of ξφ that can be obtained by several methods such as maximum likelihood (ML) or least squares, ω1 and ω2 represent weighted coefficients belonging to 0,1, ρξτ0,τ is an arbitrary loss function when ξφ is estimated by τ, and kφ is a suitable positive weight function. In this work, we discuss three types of BLF including BSE loss function, BLINEX loss function, and WBLINEX loss function, which is proposed in this work.1
Balanced Squared Error Loss Function. BSE loss function is obtained by choosing ρτ0,τ=τ−ξφ2 and kφ=1 in equation (20) as follows:(21)Lω,,τ0ξφ,τ=ω1τ−τ02+ω2τ−ξφ2,The Bayes estimation ofξφ under Lω,,τ0ξφ,τ is given by(22)τω,ξX=ω1τ0X+ω2Eξφ|X.Note thatSE loss function is a special case of BSE loss function when ω1=0.2
Balanced Linear Exponential Loss Function. The BLINEX is obtained by choosing ρτ0,τ=expcτ−ξφ−cτ−ξφ−1 and kφ=1 in equation (20) as follows:(23)Lω,τ0∗ξφ,τ=ω1expcτ−τ0−cτ−τ0−1+ω2expcτ−ξφ−cτ−ξφ−1.The Bayes estimationξφ under Lω,τ0∗ξφ,τ is given by(24)τω,ξ∗X=−1cLogω1exp−cτ0X+ω2Eexp−cξφ|X.Note that whenω1=0, then BLINEX is exactly LINEX loss function.3
Balanced Weighted Linear Exponential (BWLINEX) Loss Function. The BWLINEX is obtained by choosing(25)ρτ0,τ=exp−zφexpcτ−ξφ−cτ−ξφ−1andkφ=1,in equation (20) as follows:(26)Lω,τ0∗∗ξφ,τ=ω1exp−zφexpcτ−τ0−cτ−τ0−1+ω2exp−zφexpcτ−ξφ−cτ−ξφ−1.and the Bayes estimation ξφ under Lω,τ0∗∗ξφ,τ is given by(27)τω,ξ∗∗X=1cLogω1exp−cτ0X+ω2Eexp−zξφ|XEexp−z+cξφ|X.Note that whenω1=0, the BWLINEX loss function is exactly WLINEX loss function.


### 2.3. Bayes Estimation

In this section, we derive Bayes estimates of the scale parameterλ and the reliability Rt of the IWD. We use six different loss functions, including SE, LINEX, WLINEX, BSE, BLINEX, and BWLINEX. Under the assumption that the shape parameter θ is known, we assume a gamma (conjugate prior) for density for λ with parameters ν and η:(28)gλ=νηΓηλη−1exp−νλ,λ,ν,η>0.Combining the likelihood function in equation (5) with the prior PDFof λ in equation (28), we get the posterior of λ as(29)πλ|X=Lλ,Xgλ∫0∞Lλ,Xgλdλ=Dn+ηΓn+ηλn+η−1exp−λD,λ,ν,η>0,where X=xL1,xL2,xL3,……,xLn,(30)D=ν+Tm.


#### 2.3.1. Estimates Based on Balanced Squared Error Loss Function

Based on BSE and using equation (21), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(31)γBSE=ω1γ^ML+ω2Eγ|X,where γ^ML is the ML estimate of γ and Eγ|X can be obtained using(32)Eγ|X=∫0∞γπλ|Xdγ.Note 1. When γ=λ in equation (31), the Bayes estimation under BSE loss function of λ and denoted by λ^BSE is given by(33)λ^BSE=ω1λ^ML+ω2Eλ|X,where λ^ML is the ML estimate of λ, which can be obtained using equation (7). Eλ|X can be obtained using(34)Eλ|X=∫0∞λDn+ηΓn+ηλn+η−1exp−λDdλ=n+ηD,and D is given by equation (28).Note 2. When γ=Rt in equation (29), the Bayes estimation under BSE loss function of Rt and denoted by R^BSEt is given by(35)R^BSEt=ω1R^MLt+ω2ERt|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and ERt|X can be obtained using the following:(36)ERt|X=∫0∞1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=1−DD+t−θn+η,t≥0.The main contribution of this work is to use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BSE and R^BSEt in equations (33) and (35), respectively. To achieve this target, we minimize the MSE as follows:(37)Minimize:MSEγ^BSE=Eγ^BSE−γ2=Eω1γ^ML+ω2Eγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


#### 2.3.2. Estimates Based on BLINEX Loss Function

Based on BLINEX and using equation (24), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(38)γ^BLINEX=−1cLogω1exp−cγ^ML+ω2Eexp−cγ|X,where γ^ML is the ML estimate, and Eexp−cγ|X can be obtained using(39)Eexp−cγ|X=∫0∞exp−cγπλ|Xdγ.Note 1. When γ=λ in equation (38), the Bayes estimation under BINEX loss function of λ and denoted by γ^BLINEX is given by(40)λ^BLINEX=−1cLogω1exp−cλ^ML+ω2Eexp−cλ|X,where λ^ML is the ML estimate of λ and can be obtained using equation (7), and Eexp−cλ|X can be obtained using(41)Eexp−cλ|X=∫0∞exp−cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dc+Dn+η.Note 2. When γ=Rt in equation (38), then the Bayes estimation under BINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(42)R^BLINEXt=−1cLogω1exp−cR^MLt+ω2Eexp−c1−exp−λt−θ|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and Eexp−c1−exp−λt−θ|X can be obtained using(43)Eexp−c1−exp−λt−θ|X=∫0∞exp−c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−c+e−cDn+η∑i=1∞cii!D+it−θ−n+η.Again, we use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BLINEX and R^BLINEXt in equations (41) and (43), respectively. To achieve this target, we minimize the MSE as follows:(44)Minimize:MSEγ^BLINEX=Eγ^BLINEX−γ2=Eω1γ^ML+ω2Eexp−cγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


#### 2.3.3. Estimates Based on Weighted Balanced Loss Function

Based on WBLINEX and using equation (26), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(45)γ^WBLINEX=1cLogω1exp−cγ^ML+ω2I1I2,where γ^ML is the ML estimate of γ and I1 and I2 can be obtained by(46)I1=Eexp−zγ|X=∫0∞exp−zγπλ|Xdλ,I2=Eexp−c+zγ|X=∫0∞exp−c+zπλ|Xdλ.Note 1. When γ=λ in equation (45), the Bayes estimation under WBINEX loss function of λ and denoted by λ^WBLINEX is given by(47)λ^WBLINEX=1cLogω1exp−cλ^ML+ω2I3I4,where λ^ML is the ML estimate of λ and can be obtained using equation (7) and I3andI4 can be obtained as follows, respectively:(48)I3=Eexp−zλ|X=∫0∞exp−zλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+Dn+η,(49)I4=Eexp−z+cλ|X=∫0∞exp−z+cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+c+Dn+η.Note 2. When γ=Rt in equation (45), then the Bayes estimation under WBINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(50)R^WBLINEXt=1cLogω1exp−cR^MLt+ω2I5I6,where R^MLt is the ML estimate of Rt, which can be obtained using equation (8), and I5andI6 can be obtained as follows, respectively:(51)I5=Eexp−z1−exp−λt−θ|X=∫0∞exp−z1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+e−zDn+η∑i=1∞zii!D+it−θ−n+η,I6=Eexp−z+c1−exp−λt−θ|X=∫0∞exp−z+c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+c+e−z+cDn+η∑i=1∞z+cii!D+it−θ−n+η.As in equations (37) and (44), we minimize the MSE to compute λ^WBLINEX and R^WBLINEXt in equations (47) and (49) as follows, respectively:(52)Minimize:MSEγ^WBLINEX=Eγ^WBLINEX−γ2=Eω1γ^ML+ω2I1I2−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


### 2.4. Simulation Study and Comparisons

In this section, we conduct a Monte Carlo simulation study to compare the performance of the MLE and Bayes estimation under several loss functions, including SE, LINEX, WLINEX, BSE, BLINEX, and BWLINEX to estimate the scale parameter and reliability function of IWD when the shape parameters are known.Before beginning the simulation, we had to choose some parameters, includingc and z. We selected the values of (c) as −0.5,0.5, and 1. We selected the positive and negative values to represent both cases of upper estimate and lower estimate, respectively, while the chosen values of z are 3 and −3.We conducted the simulation according to the following steps:(1)
For the given valuesη=2,ν=1, we generated a random value λ=1.383 from the prior PDF as in equation (27).(2)
By using the valueλ=1.383 from Step 1 with selected values of θ=3, we generated data of lower record values from IWD (equation (1) for different sizes nn=3,4,5,6 and 7).(3)
We computed the estimates ofλ,Rt at a chosen time of t=0.7 using the estimations under the study.(4)
We repeated Steps 1–3 10,000 times and calculated the MSE for each estimate (sayφ) using(53)MSE=110000∑i=110000φ^i−φ2,where φ can be λ or Rt and φ^i is the estimate at the ith run.


## 2.1. Record Values and Maximum Likelihood Estimation

LetX1,X2,X3,…… be a sequence of independent and identically distributed random variables with CDF Fx and PDF fx. Set Yn=minX1,X2,X3,…,Xn, n≥1, and say that Xj is a lower record and denoted by XLi if Yj<Yj−1,j>1. Suppose we observe the first n lower record values XL1,XL2,XL3,…,XLn from the IWD whose PDF and CDF are given by (1) and (2), respectively. Based on those lower record values, we have the joint density function of the first n lower record values X≡xL1,xL2,xL3,……,xLn as given by Sultan [24]:(4)f1,2,3,……,nxL1,xL2,xL3,……,xLn=fxLn∏i=1n−1fxLi1−FxLi,0≤xL1<xL2<xL3,……<xLn<∞.Here,f. and F. are given by (1) and (2), respectively, after replacing x by xLi. The likelihood function based on the n lower record values x is given as follows:(5)ℓλ,θ|X=λθnuexp−λTm,u=∏i=1nxLi−θ−1,Tm=xLn−θ.We obtain that the log-likelihood function may be written as follows:(6)Lλ,θ|X=lnℓ=nlnλ+nlnθ−λTm−θ+1∑i=1nlnxLi.Assuming that the shape parameterθ is known, using equation (6), the maximum likelihood estimator (MLE) λ^ML of the scale parameter λ can be shown to be of the following form:(7)λ^ML=nxLn−θ.Ifλ is replaced by λ^ML in equation (3), we can obtain the MLE of reliability function R^MLt of R (t) depending on the invariance property:(8)R^MLt=1−exp−λ^MLt−θ.


## 2.2. Loss Functions

In the following sections, we present the four main types of loss functions under investigation in this work.


### 2.2.1. Squared Error Loss Function

The SE loss function can be expressed as(9)Lφ^,φ=φ^−φ2.The Bayes estimator ofφ based on SE loss function can be obtained as follows:(10)φ^SE=Eπφ|X.


### 2.2.2. Linear Exponential Loss Function

Varian [25] introduced the LINEX loss function. LINEX is an asymmetric loss function that can be expressed as(11)LΔ∝expcΔ−cΔ−1,c≠0,where Δ=φ^−φ. The sign and magnitude of c reflect the direction and degree of asymmetry, respectively. The Bayes estimator relative to LINEX loss function, denoted by φ^LINEX, is given as follows:(12)φ^LINEX=−1cLnEφexp−cφ,c≠0,provided that Eφ=e−cφ exists and is finite, where Eφ denotes the expected value.


### 2.2.3. Weighted Linear Exponential Loss Functions

The researcher proposes this loss function depending on WLINEX loss function as follows:(13)L⋆Δ=L⋆φ^−φ∝wφexpcΔ−cΔ−1,c≠0.Here,φ^ represents the estimated parameter that makes the expectation of loss function by equation (13) as small as possible. The value wφ represents the proposed weighted function, which is equal to the following:(14)wφ=exp−zφ.Depending on the posterior distribution of the parameterφ and using the proposed weighted function as in equation (14), we can attain the estimated weighted Bayes of the parameter φ as follows:(15)ELw⋆φ^,φ=∫∀φLwφ^,φfφ|Xdφ=∫∀φwφexpcφ^−φ−cφ^−φ−1fφ|Xdφ=∫∀φexp−zφexpcφ^−φfφ|Xdφ−∫∀φexp−zφcφ^−φfφ|Xdφ−∫∀φexp−zφfφ|Xdφ=expcφ^∫∀φexp−φz+cfφ|Xdφ−cφ^∫∀φexp−zφfφ|Xdφ+c∫∀φφexp−zφfφ|Xdφ−∫∀φexp−zφfφ|Xdφexpcφ^Eφexp−φz+c|X−cφ^Eφexp−zφ|X+cEφφexp−zφ|X−Eφexp−zφ|X.It is known that, to find the value ofφ^ that minimizes ELw⋆φ^,φ, we have to perform the following two steps:(i)∂EφLw⋆φ^,φ/∂φ^=0(16)∂EφLw⋆φ^,φ∂φ^=cexpcφ^Eφexp−φz+c|X−cEφexp−zφ|X=0.Therefore, we can find the following:(17)c=expcφ^Eφexp−φz+c|X=cEφexp−zφ|X.Consequently, the Bayesian estimation of the parameterφ using WLINEX will be(18)φ^WLINEX=1cLnEφexp−zφ|XEφexp−φz+c|X,z+c≠0.(i)
Eφ=e−zφ and Eφ=e−z+cφ exist and are finite, where Eφ denotes the expected value.(ii
)∂2EφLw⋆φ^,φ/∂φ^2>0 at the minimum value computed by (i):(19)∂2EφLw⋆φ^,φ∂φ^2=∂∂φ^∂EφLw⋆φ^,φ∂φ^=c2expcφ^Eφexp−φz+c>0.Becauseφ^WLINEX satisfies conditions (i) and (ii), it follows that φ^WLINEX is the minimum value. Note that the WLINEX loss function is a generalization of the LINEX loss function, where LINEX is a special case of WLINEX when z=0 in equation (18).


### 2.2.4. Balanced Loss Function

According to AbdEllah [22], the class of balanced loss function (BLF) can be written in the form(20)Lρ,ω,τ0kξφ,τ=ω1kφρτ0,τ+ω2kφρξφ,τ,ω1+ω2=1,where τ represents an estimator of parameter ξφ,τ0 is a chosen prior estimator of ξφ that can be obtained by several methods such as maximum likelihood (ML) or least squares, ω1 and ω2 represent weighted coefficients belonging to 0,1, ρξτ0,τ is an arbitrary loss function when ξφ is estimated by τ, and kφ is a suitable positive weight function. In this work, we discuss three types of BLF including BSE loss function, BLINEX loss function, and WBLINEX loss function, which is proposed in this work.1
Balanced Squared Error Loss Function. BSE loss function is obtained by choosing ρτ0,τ=τ−ξφ2 and kφ=1 in equation (20) as follows:(21)Lω,,τ0ξφ,τ=ω1τ−τ02+ω2τ−ξφ2,The Bayes estimation ofξφ under Lω,,τ0ξφ,τ is given by(22)τω,ξX=ω1τ0X+ω2Eξφ|X.Note thatSE loss function is a special case of BSE loss function when ω1=0.2
Balanced Linear Exponential Loss Function. The BLINEX is obtained by choosing ρτ0,τ=expcτ−ξφ−cτ−ξφ−1 and kφ=1 in equation (20) as follows:(23)Lω,τ0∗ξφ,τ=ω1expcτ−τ0−cτ−τ0−1+ω2expcτ−ξφ−cτ−ξφ−1.The Bayes estimationξφ under Lω,τ0∗ξφ,τ is given by(24)τω,ξ∗X=−1cLogω1exp−cτ0X+ω2Eexp−cξφ|X.Note that whenω1=0, then BLINEX is exactly LINEX loss function.3
Balanced Weighted Linear Exponential (BWLINEX) Loss Function. The BWLINEX is obtained by choosing(25)ρτ0,τ=exp−zφexpcτ−ξφ−cτ−ξφ−1andkφ=1,in equation (20) as follows:(26)Lω,τ0∗∗ξφ,τ=ω1exp−zφexpcτ−τ0−cτ−τ0−1+ω2exp−zφexpcτ−ξφ−cτ−ξφ−1.and the Bayes estimation ξφ under Lω,τ0∗∗ξφ,τ is given by(27)τω,ξ∗∗X=1cLogω1exp−cτ0X+ω2Eexp−zξφ|XEexp−z+cξφ|X.Note that whenω1=0, the BWLINEX loss function is exactly WLINEX loss function.


## 2.2.1. Squared Error Loss Function

The SE loss function can be expressed as(9)Lφ^,φ=φ^−φ2.The Bayes estimator ofφ based on SE loss function can be obtained as follows:(10)φ^SE=Eπφ|X.


## 2.2.2. Linear Exponential Loss Function

Varian [25] introduced the LINEX loss function. LINEX is an asymmetric loss function that can be expressed as(11)LΔ∝expcΔ−cΔ−1,c≠0,where Δ=φ^−φ. The sign and magnitude of c reflect the direction and degree of asymmetry, respectively. The Bayes estimator relative to LINEX loss function, denoted by φ^LINEX, is given as follows:(12)φ^LINEX=−1cLnEφexp−cφ,c≠0,provided that Eφ=e−cφ exists and is finite, where Eφ denotes the expected value.


## 2.2.3. Weighted Linear Exponential Loss Functions

The researcher proposes this loss function depending on WLINEX loss function as follows:(13)L⋆Δ=L⋆φ^−φ∝wφexpcΔ−cΔ−1,c≠0.Here,φ^ represents the estimated parameter that makes the expectation of loss function by equation (13) as small as possible. The value wφ represents the proposed weighted function, which is equal to the following:(14)wφ=exp−zφ.Depending on the posterior distribution of the parameterφ and using the proposed weighted function as in equation (14), we can attain the estimated weighted Bayes of the parameter φ as follows:(15)ELw⋆φ^,φ=∫∀φLwφ^,φfφ|Xdφ=∫∀φwφexpcφ^−φ−cφ^−φ−1fφ|Xdφ=∫∀φexp−zφexpcφ^−φfφ|Xdφ−∫∀φexp−zφcφ^−φfφ|Xdφ−∫∀φexp−zφfφ|Xdφ=expcφ^∫∀φexp−φz+cfφ|Xdφ−cφ^∫∀φexp−zφfφ|Xdφ+c∫∀φφexp−zφfφ|Xdφ−∫∀φexp−zφfφ|Xdφexpcφ^Eφexp−φz+c|X−cφ^Eφexp−zφ|X+cEφφexp−zφ|X−Eφexp−zφ|X.It is known that, to find the value ofφ^ that minimizes ELw⋆φ^,φ, we have to perform the following two steps:(i)∂EφLw⋆φ^,φ/∂φ^=0(16)∂EφLw⋆φ^,φ∂φ^=cexpcφ^Eφexp−φz+c|X−cEφexp−zφ|X=0.Therefore, we can find the following:(17)c=expcφ^Eφexp−φz+c|X=cEφexp−zφ|X.Consequently, the Bayesian estimation of the parameterφ using WLINEX will be(18)φ^WLINEX=1cLnEφexp−zφ|XEφexp−φz+c|X,z+c≠0.(i)
Eφ=e−zφ and Eφ=e−z+cφ exist and are finite, where Eφ denotes the expected value.(ii
)∂2EφLw⋆φ^,φ/∂φ^2>0 at the minimum value computed by (i):(19)∂2EφLw⋆φ^,φ∂φ^2=∂∂φ^∂EφLw⋆φ^,φ∂φ^=c2expcφ^Eφexp−φz+c>0.Becauseφ^WLINEX satisfies conditions (i) and (ii), it follows that φ^WLINEX is the minimum value. Note that the WLINEX loss function is a generalization of the LINEX loss function, where LINEX is a special case of WLINEX when z=0 in equation (18).


## 2.2.4. Balanced Loss Function

According to AbdEllah [22], the class of balanced loss function (BLF) can be written in the form(20)Lρ,ω,τ0kξφ,τ=ω1kφρτ0,τ+ω2kφρξφ,τ,ω1+ω2=1,where τ represents an estimator of parameter ξφ,τ0 is a chosen prior estimator of ξφ that can be obtained by several methods such as maximum likelihood (ML) or least squares, ω1 and ω2 represent weighted coefficients belonging to 0,1, ρξτ0,τ is an arbitrary loss function when ξφ is estimated by τ, and kφ is a suitable positive weight function. In this work, we discuss three types of BLF including BSE loss function, BLINEX loss function, and WBLINEX loss function, which is proposed in this work.1
Balanced Squared Error Loss Function. BSE loss function is obtained by choosing ρτ0,τ=τ−ξφ2 and kφ=1 in equation (20) as follows:(21)Lω,,τ0ξφ,τ=ω1τ−τ02+ω2τ−ξφ2,The Bayes estimation ofξφ under Lω,,τ0ξφ,τ is given by(22)τω,ξX=ω1τ0X+ω2Eξφ|X.Note thatSE loss function is a special case of BSE loss function when ω1=0.2
Balanced Linear Exponential Loss Function. The BLINEX is obtained by choosing ρτ0,τ=expcτ−ξφ−cτ−ξφ−1 and kφ=1 in equation (20) as follows:(23)Lω,τ0∗ξφ,τ=ω1expcτ−τ0−cτ−τ0−1+ω2expcτ−ξφ−cτ−ξφ−1.The Bayes estimationξφ under Lω,τ0∗ξφ,τ is given by(24)τω,ξ∗X=−1cLogω1exp−cτ0X+ω2Eexp−cξφ|X.Note that whenω1=0, then BLINEX is exactly LINEX loss function.3
Balanced Weighted Linear Exponential (BWLINEX) Loss Function. The BWLINEX is obtained by choosing(25)ρτ0,τ=exp−zφexpcτ−ξφ−cτ−ξφ−1andkφ=1,in equation (20) as follows:(26)Lω,τ0∗∗ξφ,τ=ω1exp−zφexpcτ−τ0−cτ−τ0−1+ω2exp−zφexpcτ−ξφ−cτ−ξφ−1.and the Bayes estimation ξφ under Lω,τ0∗∗ξφ,τ is given by(27)τω,ξ∗∗X=1cLogω1exp−cτ0X+ω2Eexp−zξφ|XEexp−z+cξφ|X.Note that whenω1=0, the BWLINEX loss function is exactly WLINEX loss function.


## 2.3. Bayes Estimation

In this section, we derive Bayes estimates of the scale parameterλ and the reliability Rt of the IWD. We use six different loss functions, including SE, LINEX, WLINEX, BSE, BLINEX, and BWLINEX. Under the assumption that the shape parameter θ is known, we assume a gamma (conjugate prior) for density for λ with parameters ν and η:(28)gλ=νηΓηλη−1exp−νλ,λ,ν,η>0.Combining the likelihood function in equation (5) with the prior PDFof λ in equation (28), we get the posterior of λ as(29)πλ|X=Lλ,Xgλ∫0∞Lλ,Xgλdλ=Dn+ηΓn+ηλn+η−1exp−λD,λ,ν,η>0,where X=xL1,xL2,xL3,……,xLn,(30)D=ν+Tm.


### 2.3.1. Estimates Based on Balanced Squared Error Loss Function

Based on BSE and using equation (21), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(31)γBSE=ω1γ^ML+ω2Eγ|X,where γ^ML is the ML estimate of γ and Eγ|X can be obtained using(32)Eγ|X=∫0∞γπλ|Xdγ.Note 1. When γ=λ in equation (31), the Bayes estimation under BSE loss function of λ and denoted by λ^BSE is given by(33)λ^BSE=ω1λ^ML+ω2Eλ|X,where λ^ML is the ML estimate of λ, which can be obtained using equation (7). Eλ|X can be obtained using(34)Eλ|X=∫0∞λDn+ηΓn+ηλn+η−1exp−λDdλ=n+ηD,and D is given by equation (28).Note 2. When γ=Rt in equation (29), the Bayes estimation under BSE loss function of Rt and denoted by R^BSEt is given by(35)R^BSEt=ω1R^MLt+ω2ERt|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and ERt|X can be obtained using the following:(36)ERt|X=∫0∞1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=1−DD+t−θn+η,t≥0.The main contribution of this work is to use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BSE and R^BSEt in equations (33) and (35), respectively. To achieve this target, we minimize the MSE as follows:(37)Minimize:MSEγ^BSE=Eγ^BSE−γ2=Eω1γ^ML+ω2Eγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


### 2.3.2. Estimates Based on BLINEX Loss Function

Based on BLINEX and using equation (24), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(38)γ^BLINEX=−1cLogω1exp−cγ^ML+ω2Eexp−cγ|X,where γ^ML is the ML estimate, and Eexp−cγ|X can be obtained using(39)Eexp−cγ|X=∫0∞exp−cγπλ|Xdγ.Note 1. When γ=λ in equation (38), the Bayes estimation under BINEX loss function of λ and denoted by γ^BLINEX is given by(40)λ^BLINEX=−1cLogω1exp−cλ^ML+ω2Eexp−cλ|X,where λ^ML is the ML estimate of λ and can be obtained using equation (7), and Eexp−cλ|X can be obtained using(41)Eexp−cλ|X=∫0∞exp−cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dc+Dn+η.Note 2. When γ=Rt in equation (38), then the Bayes estimation under BINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(42)R^BLINEXt=−1cLogω1exp−cR^MLt+ω2Eexp−c1−exp−λt−θ|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and Eexp−c1−exp−λt−θ|X can be obtained using(43)Eexp−c1−exp−λt−θ|X=∫0∞exp−c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−c+e−cDn+η∑i=1∞cii!D+it−θ−n+η.Again, we use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BLINEX and R^BLINEXt in equations (41) and (43), respectively. To achieve this target, we minimize the MSE as follows:(44)Minimize:MSEγ^BLINEX=Eγ^BLINEX−γ2=Eω1γ^ML+ω2Eexp−cγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


### 2.3.3. Estimates Based on Weighted Balanced Loss Function

Based on WBLINEX and using equation (26), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(45)γ^WBLINEX=1cLogω1exp−cγ^ML+ω2I1I2,where γ^ML is the ML estimate of γ and I1 and I2 can be obtained by(46)I1=Eexp−zγ|X=∫0∞exp−zγπλ|Xdλ,I2=Eexp−c+zγ|X=∫0∞exp−c+zπλ|Xdλ.Note 1. When γ=λ in equation (45), the Bayes estimation under WBINEX loss function of λ and denoted by λ^WBLINEX is given by(47)λ^WBLINEX=1cLogω1exp−cλ^ML+ω2I3I4,where λ^ML is the ML estimate of λ and can be obtained using equation (7) and I3andI4 can be obtained as follows, respectively:(48)I3=Eexp−zλ|X=∫0∞exp−zλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+Dn+η,(49)I4=Eexp−z+cλ|X=∫0∞exp−z+cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+c+Dn+η.Note 2. When γ=Rt in equation (45), then the Bayes estimation under WBINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(50)R^WBLINEXt=1cLogω1exp−cR^MLt+ω2I5I6,where R^MLt is the ML estimate of Rt, which can be obtained using equation (8), and I5andI6 can be obtained as follows, respectively:(51)I5=Eexp−z1−exp−λt−θ|X=∫0∞exp−z1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+e−zDn+η∑i=1∞zii!D+it−θ−n+η,I6=Eexp−z+c1−exp−λt−θ|X=∫0∞exp−z+c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+c+e−z+cDn+η∑i=1∞z+cii!D+it−θ−n+η.As in equations (37) and (44), we minimize the MSE to compute λ^WBLINEX and R^WBLINEXt in equations (47) and (49) as follows, respectively:(52)Minimize:MSEγ^WBLINEX=Eγ^WBLINEX−γ2=Eω1γ^ML+ω2I1I2−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


## 2.3.1. Estimates Based on Balanced Squared Error Loss Function

Based on BSE and using equation (21), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(31)γBSE=ω1γ^ML+ω2Eγ|X,where γ^ML is the ML estimate of γ and Eγ|X can be obtained using(32)Eγ|X=∫0∞γπλ|Xdγ.Note 1. When γ=λ in equation (31), the Bayes estimation under BSE loss function of λ and denoted by λ^BSE is given by(33)λ^BSE=ω1λ^ML+ω2Eλ|X,where λ^ML is the ML estimate of λ, which can be obtained using equation (7). Eλ|X can be obtained using(34)Eλ|X=∫0∞λDn+ηΓn+ηλn+η−1exp−λDdλ=n+ηD,and D is given by equation (28).Note 2. When γ=Rt in equation (29), the Bayes estimation under BSE loss function of Rt and denoted by R^BSEt is given by(35)R^BSEt=ω1R^MLt+ω2ERt|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and ERt|X can be obtained using the following:(36)ERt|X=∫0∞1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=1−DD+t−θn+η,t≥0.The main contribution of this work is to use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BSE and R^BSEt in equations (33) and (35), respectively. To achieve this target, we minimize the MSE as follows:(37)Minimize:MSEγ^BSE=Eγ^BSE−γ2=Eω1γ^ML+ω2Eγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


## 2.3.2. Estimates Based on BLINEX Loss Function

Based on BLINEX and using equation (24), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(38)γ^BLINEX=−1cLogω1exp−cγ^ML+ω2Eexp−cγ|X,where γ^ML is the ML estimate, and Eexp−cγ|X can be obtained using(39)Eexp−cγ|X=∫0∞exp−cγπλ|Xdγ.Note 1. When γ=λ in equation (38), the Bayes estimation under BINEX loss function of λ and denoted by γ^BLINEX is given by(40)λ^BLINEX=−1cLogω1exp−cλ^ML+ω2Eexp−cλ|X,where λ^ML is the ML estimate of λ and can be obtained using equation (7), and Eexp−cλ|X can be obtained using(41)Eexp−cλ|X=∫0∞exp−cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dc+Dn+η.Note 2. When γ=Rt in equation (38), then the Bayes estimation under BINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(42)R^BLINEXt=−1cLogω1exp−cR^MLt+ω2Eexp−c1−exp−λt−θ|X,where R^MLt is the ML estimate of Rt and can be obtained using equation (8), and Eexp−c1−exp−λt−θ|X can be obtained using(43)Eexp−c1−exp−λt−θ|X=∫0∞exp−c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−c+e−cDn+η∑i=1∞cii!D+it−θ−n+η.Again, we use nonlinear programming to find the optimal values ofω1 and ω2 to compute λ^BLINEX and R^BLINEXt in equations (41) and (43), respectively. To achieve this target, we minimize the MSE as follows:(44)Minimize:MSEγ^BLINEX=Eγ^BLINEX−γ2=Eω1γ^ML+ω2Eexp−cγ|X−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


## 2.3.3. Estimates Based on Weighted Balanced Loss Function

Based on WBLINEX and using equation (26), the Bayes estimation of a parameter γ (which can be the scale parameter λ or the reliability function Rt) is given by(45)γ^WBLINEX=1cLogω1exp−cγ^ML+ω2I1I2,where γ^ML is the ML estimate of γ and I1 and I2 can be obtained by(46)I1=Eexp−zγ|X=∫0∞exp−zγπλ|Xdλ,I2=Eexp−c+zγ|X=∫0∞exp−c+zπλ|Xdλ.Note 1. When γ=λ in equation (45), the Bayes estimation under WBINEX loss function of λ and denoted by λ^WBLINEX is given by(47)λ^WBLINEX=1cLogω1exp−cλ^ML+ω2I3I4,where λ^ML is the ML estimate of λ and can be obtained using equation (7) and I3andI4 can be obtained as follows, respectively:(48)I3=Eexp−zλ|X=∫0∞exp−zλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+Dn+η,(49)I4=Eexp−z+cλ|X=∫0∞exp−z+cλDn+ηΓn+ηλn+η−1exp−λDdλ=Dz+c+Dn+η.Note 2. When γ=Rt in equation (45), then the Bayes estimation under WBINEX loss function of Rt, which is denoted by R^BLINEXt, is given by(50)R^WBLINEXt=1cLogω1exp−cR^MLt+ω2I5I6,where R^MLt is the ML estimate of Rt, which can be obtained using equation (8), and I5andI6 can be obtained as follows, respectively:(51)I5=Eexp−z1−exp−λt−θ|X=∫0∞exp−z1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+e−zDn+η∑i=1∞zii!D+it−θ−n+η,I6=Eexp−z+c1−exp−λt−θ|X=∫0∞exp−z+c1−exp−λt−θDn+ηΓn+ηλn+η−1exp−λDdλ=e−z+c+e−z+cDn+η∑i=1∞z+cii!D+it−θ−n+η.As in equations (37) and (44), we minimize the MSE to compute λ^WBLINEX and R^WBLINEXt in equations (47) and (49) as follows, respectively:(52)Minimize:MSEγ^WBLINEX=Eγ^WBLINEX−γ2=Eω1γ^ML+ω2I1I2−γ2subject toω1+ω2=1,0≤ω1<1,0≤ω2<1.


## 2.4. Simulation Study and Comparisons

In this section, we conduct a Monte Carlo simulation study to compare the performance of the MLE and Bayes estimation under several loss functions, including SE, LINEX, WLINEX, BSE, BLINEX, and BWLINEX to estimate the scale parameter and reliability function of IWD when the shape parameters are known.Before beginning the simulation, we had to choose some parameters, includingc and z. We selected the values of (c) as −0.5,0.5, and 1. We selected the positive and negative values to represent both cases of upper estimate and lower estimate, respectively, while the chosen values of z are 3 and −3.We conducted the simulation according to the following steps:(1)
For the given valuesη=2,ν=1, we generated a random value λ=1.383 from the prior PDF as in equation (27).(2)
By using the valueλ=1.383 from Step 1 with selected values of θ=3, we generated data of lower record values from IWD (equation (1) for different sizes nn=3,4,5,6 and 7).(3)
We computed the estimates ofλ,Rt at a chosen time of t=0.7 using the estimations under the study.(4)
We repeated Steps 1–3 10,000 times and calculated the MSE for each estimate (sayφ) using(53)MSE=110000∑i=110000φ^i−φ2,where φ can be λ or Rt and φ^i is the estimate at the ith run.


## 3. Results and Discussion

The results of the simulation are listed in Tables1–4.Table 1
MSEs of the estimates ofλ under unbalanced loss functions at z=3.
nλ^MLλ^SEλ^LINEXλ^WLINEXc=−0.5c=0.5c=1c=−0.5c=0.5c=134.740990.673991.231710.409320.270930.280150.350970.3877341.792420.545690.892380.388130.269380.232600.288000.3163951.127250.492520.746560.335400.245350.193470.238690.2640960.794180.420790.604210.296100.237870.163850.206250.2222170.523760.337020.463910.257160.191960.139000.171250.18800Table 2
MSEs of the estimates ofλ under balanced loss functions at z=3.
nλ^BSEλ^BLINEXλ^WBLINEXc=−0.5c=0.5c=1c=−0.5c=0.5c=130.658111.131770.408290.270920.280140.350970.3877340.532060.810090.387250.269380.232570.288000.3163950.480360.676640.334630.245350.193000.238670.2640960.409740.542450.295410.237860.162360.206060.2221370.327080.410680.256490.191960.136640.170690.18782Table 3
MSEs of the estimates ofRt under unbalanced loss functions at z=−3.
nR^MLtR^SEtR^LINEXtR^WLINEXtc=−0.5c=0.5c=1c=−0.5c=0.5c=130.003010.002150.002020.002290.002410.001030.001130.0011840.002020.001680.001600.001720.001880.000890.000940.0010250.001400.001330.001270.001390.001540.000760.000810.0009060.000910.000960.000920.001040.001060.000580.000650.0006570.000580.000680.000660.000730.000760.000430.000470.00049Table 4
MSEs of the estimates ofRt under balanced loss functions at z=−3.
nR^BSEtR^BLINEXtR^BWLINEXtc=−0.5c=0.5c=1c=−0.5c=0.5c=130.001910.002020.002290.002410.000990.001100.0011540.001480.001600.001720.001880.000850.000910.0009950.001140.001270.001390.001540.000720.000780.0008660.000780.000920.001040.001060.000540.000610.0006270.000510.000650.000730.000750.000390.000440.00045In this paper, we employed nonlinear programming to obtain the best values of weighted coefficients (ω1 and ω2) of the balanced loss function. The estimates of the parameter λ and reliability function R (t) follow the IWD. The estimation methods under study include ML, SE, LINEX, WLINEX, BSE, BINEX, and BWLINEX. We conducted the estimations depending on lower record values.The main observations of the results are stated in the following points:(1)
Tables1–4 reveal that the proposed estimator BWLINEX is the absolute best method to estimate scale parameter λ and reliability Rt according to the smallest values of MSEs compared with the estimators of all other methods under study.(2)
For unbalanced loss functions, Tables1 and 3 showed that the proposed estimator WLINEX is the best method to estimate scale parameter λ and reliability Rt according to the smallest values of MSEs compared with the estimators of ML, SE, and LINEX.(3)
The results show that the values of all MSEs decrease asn increases. This means that there is an inverse relationship between the evaluation functions and the number of recorded values.(4)
It is clear from Table1 that the ML method has the worst performance in estimating the scale parameter λ, especially at the small-size samples.(5)
The results show that estimators of the different methods are closer to each other as the sample size increases.(6)
Tables1 and 2 show that the lower estimate (c=−0.5) of the scale parameter λ is better than the upper estimate (c=0.5 and c=1) when estimating by WLINEX. However, the upper estimate is better than the lower estimate when estimating by LINEX.(7)
The Bayesian estimators relative to LINEX, WLINEX, and BWLINEX are sensitive to the values of the corresponding shape parameterc.(8)
The balanced Bayesian estimator’s performance is consistently better than the corresponding unbalanced Bayesian estimator.


## 4. Conclusion

In this work, we developed LINEX to estimate the scale parameter and reliability function of IWD depending on lower record values. The development occurred through merging a weight into LINEX to produce a new loss function called WLINEX. We used WLINEX to derive the scale parameter and reliability function of the IWD. The majority of earlier researchers have determined the weighted balanced coefficients by arbitrarily providing the summation equal to one. In this work, we depended on justifiable mathematical methods to determine these coefficients, where we utilized nonlinear programming to obtain the optimal values of the weighted coefficients for each of BSE, BLINEX, and BWLINEX.Furthermore, we conducted a Monte Carlo simulation to examine the performance of the proposed methods: WLINEX and BWLINEX. We then compared the proposed methods with the other methods, including ML, SE, LINEX, BSE, and BLINEX. The results of the simulation showed that the developed estimators in this work (BWLINEX and WLINEX) have the best performance in estimating scale parameter and in estimating reliability according to the smallest values of MSE, respectively. This result shows that the approach followed is promising and can be applied in a real environment.



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*Source: 6648462-2021-03-25.xml*