# Bayesian Estimations under the Weighted LINEX Loss Function Based on Upper Record Values

Authors: Fuad S. Al-Duais
Journal: Complexity
 (2021)

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## Abstract

The essential objective of this research is to develop a linear exponential (LINEX) loss function to estimate the parameters and reliability function of the Weibull distribution (WD) based on upper record values when both shape and scale parameters are unknown. We perform this by merging a weight into LINEX to produce a new loss function called the weighted linear exponential (WLINEX) loss function. Then, we utilized WLINEX to derive the parameters and reliability function of the WD. Next, we compared the performance of the proposed method (WLINEX) in this work with Bayesian estimation using the LINEX loss function, Bayesian estimation using the squared-error (SEL) loss function, and maximum likelihood estimation (MLE). The evaluation depended on the difference between the estimated parameters and the parameters of completed data. The results revealed that the proposed method is the best for estimating parameters and has good performance for estimating reliability.


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## Body

## 1. Introduction

Record values appeared naturally in many real-life applications such as data relating to sport, weather, and life-testing studies. Many authors studied record values and the associated statistics such as [1–6]. Furthermore, there are several studies which discussed some inferential methods based on record values for the Rayleigh [7, 8], Weibull [9], inverse Weibull [10, 11], exponentiated Family [12, 13], Lomax [14, 15], power Lindley model [16], exponentiated Weibull [17, 18], and normal distribution [19].In this paper, we derive the Bayes estimator under the weighted linear exponential (WLINEX) loss function to obtain parameters and the reliability function of the Weibull distribution based on record values. Afterwards, we compare the proposed model with others.LetX1,X2,X3,… be a sequence of independent and identically distributed (iid) random variables with (c.d.f) Fx and (p.d.f) fx. Set Ym=maxX1,X2,X3,…,Xm, m≥1, and we say that Xj is an upper record and denoted by XUi if Yj>Yj−1,j>1.Assume thatXU1,XU2,XU3,…,XUm are the first m-upper record values arising from a sequence Xi of (iid) Weibull variables with probability density function (p.d.f)(1)fx;α,β=αβxα−1exp−βxα,x≥0,α,β>0,0,o.w,and cumulative distribution function (c.d.f)(2)Fx;α,β=1−exp−βxα,x≥0,α,β>0,where β and α are scale and shape parameters, respectively. Additionally, the reliability function R(t) and the hazard (instantaneous failure rate) function H(t) at mission time t for the Weibull distribution are given by(3)Rt;α,β=1−FTt;α,β=exp−βtα,t≥0,(4)Ht=αβtα−1,t≥0.


## 2. Maximum Likelihood Estimates (MLEs)

In this section, we discuss the maximum likelihood estimates of the parameters of WD given in (1) when the available data are record values.Suppose we observe the firstm-upper record values, each of which has the Weibull distribution whose p.d.f and c.d.f are given by (1) and (2), respectively. Based on these upper record values, for simplicity of notation, we will use xi instead of XUi. We have the joint density function of the first m-upper record values x≡xU1,xU2,xU3,…,xUm which is given according to Arnold et al. [3] by(5)f1,2,3,…,mXU1,XU2,XU3,…,XUm=fxUm∏i=1m−1fxUi1−FxUi,0≤XU1<XU2<XU3,…,<XUm<∞,where f·, and F· are given, by (1) and (2), respectively, after replacing x by xUi.The likelihood function based on them-upper record values x is given as follows:(6)ℓα,βx=αβmuexp−βTm,u=∏i=1mxUiα−1,Tm=xUmα.From (6), the natural logarithm of the likelihood function is given by(7)Lα,β|x≡Lnℓ=mLnαβ−βTm+α−1∑i=1mLnxUi,where Tm is given by equation (6).When bothα and β are unknown, the maximum likelihood estimates of β^ML and α^ML can be obtained from (7) by solving the following two equations:(8)α^ML=mmLnxUm−∑i=1mLnxUi,(9)β^ML=mxUmα^ML.By the invariance property of the MLE, reliability function−R^tML of Rt can be obtained as given by (3) after replacing α^ML and β^ML by:(10)R^tML=exp−β^MLtα^ML,t≥0.


## 3. Loss Function

We consider three different loss functions.


### 3.1. Squared-Error Loss Function (SEL)

(11)Lθ^,θ=θ^−θ2.The squared-error loss function (quadratic loss) is classified as a symmetric function and associates equal importance to the losses for overestimation and underestimation of equal magnitude. The Bayes estimateθ^BS under the above loss function is the posterior mean, i.e., θ^Bs=Eθ.


### 3.2. LINEX Loss Function

The LINEX loss function forθ can be expressed as [20](12)LΔ∝expaΔ−aΔ−1,a≠0,where Δ=θ^−θ. Here, a represents the shape parameter of the loss function. The behavior of the LINEX loss function changes with the choice of a. In particular, if a is close to zero, this loss function is close to the SEL loss function and therefore almost symmetric.The Bayes estimator ofθ denoted byθ^BL, depending on the LINEX loss function, is given by(13)θ^BL=−1alnΕθexp−aθ,a≠0,provided that Eθ=e−aθ exists and is finite, where Eθ denotes the expected value.


### 3.3. Weighted LINEX Loss Function (WLINEX)

This function is proposed by the researcher depending on the weighted LINEX loss function (WLINEX) as follows:(14)Lwθ^−θ=wθexpaΔ−aΔ−1,a≠0,where θ^ represents the estimated parameter that makes the expectation of the loss function (equation (14)) as small as possible. The value wθ represents the proposed weighted function, which is given as follows:(15)wθ=exp−zθ.Depending on the posterior distribution of parameterθ and by using the proposed weighted function, we get the Bayesian estimation of parameter θ as follows:(16)ΕLwθ^,θ=∫∀θLwθ^,θfθ|xdθ=∫∀θwθexpaθ^−θ−aθ^−θ−1fθ|xdθ=∫∀θexp−zθexpaθ^−θfθ|xdθ−∫∀θexp−zθaθ^−θfθ|xdθ−∫∀θexp−zθfθ|xdθ=expaθ^∫∀θexp−θa+zfθ|xdθ−aθ^∫∀θexp−zθfθ|xdθ+a∫∀θθexp−zθfθ|xdθ−∫∀θexp−zθfθ|xdθ=expaθ^Eθexp−θz+a|x−aθ^Eθexp−zθ|x+aEθθexp−zθ|x−Eexp−zθ|x=∂EθLwθ^,θ∂θ^=aexpaθ^Eexp−θz+a|x−aEθexp−zθ|x=0.So, we can find that(17)aexpaθ^Eexp−θz+a|x−=aEexp−zθ|x.Consequently, the Bayesian estimation of parameterθ using WLINEX will be(18)θ^WBL=1aLnEexp−zθ−1aLnEexp−θz+a=1aLnEexp−zθEexp−θz+a.


## 3.1. Squared-Error Loss Function (SEL)

(11)Lθ^,θ=θ^−θ2.The squared-error loss function (quadratic loss) is classified as a symmetric function and associates equal importance to the losses for overestimation and underestimation of equal magnitude. The Bayes estimateθ^BS under the above loss function is the posterior mean, i.e., θ^Bs=Eθ.


## 3.2. LINEX Loss Function

The LINEX loss function forθ can be expressed as [20](12)LΔ∝expaΔ−aΔ−1,a≠0,where Δ=θ^−θ. Here, a represents the shape parameter of the loss function. The behavior of the LINEX loss function changes with the choice of a. In particular, if a is close to zero, this loss function is close to the SEL loss function and therefore almost symmetric.The Bayes estimator ofθ denoted byθ^BL, depending on the LINEX loss function, is given by(13)θ^BL=−1alnΕθexp−aθ,a≠0,provided that Eθ=e−aθ exists and is finite, where Eθ denotes the expected value.


## 3.3. Weighted LINEX Loss Function (WLINEX)

This function is proposed by the researcher depending on the weighted LINEX loss function (WLINEX) as follows:(14)Lwθ^−θ=wθexpaΔ−aΔ−1,a≠0,where θ^ represents the estimated parameter that makes the expectation of the loss function (equation (14)) as small as possible. The value wθ represents the proposed weighted function, which is given as follows:(15)wθ=exp−zθ.Depending on the posterior distribution of parameterθ and by using the proposed weighted function, we get the Bayesian estimation of parameter θ as follows:(16)ΕLwθ^,θ=∫∀θLwθ^,θfθ|xdθ=∫∀θwθexpaθ^−θ−aθ^−θ−1fθ|xdθ=∫∀θexp−zθexpaθ^−θfθ|xdθ−∫∀θexp−zθaθ^−θfθ|xdθ−∫∀θexp−zθfθ|xdθ=expaθ^∫∀θexp−θa+zfθ|xdθ−aθ^∫∀θexp−zθfθ|xdθ+a∫∀θθexp−zθfθ|xdθ−∫∀θexp−zθfθ|xdθ=expaθ^Eθexp−θz+a|x−aθ^Eθexp−zθ|x+aEθθexp−zθ|x−Eexp−zθ|x=∂EθLwθ^,θ∂θ^=aexpaθ^Eexp−θz+a|x−aEθexp−zθ|x=0.So, we can find that(17)aexpaθ^Eexp−θz+a|x−=aEexp−zθ|x.Consequently, the Bayesian estimation of parameterθ using WLINEX will be(18)θ^WBL=1aLnEexp−zθ−1aLnEexp−θz+a=1aLnEexp−zθEexp−θz+a.


## 4. Bayes Estimation

In the case of the two-parameter problem, we need to specify a general joint prior forα and β which may lead to complicated calculations. Aiming to solve this problem, Soland’s method was used. Soland [21] considered a family of joint prior distributions that places continuous distributions on the scale parameter and discrete distributions on the shape parameter.Suppose thatα is restricted to the values α1,α2,α3,…,αk with prior probabilities η1,η2,η3,…,ηk, that is,(19)Pαj=Prα=αj=ηj,j=1,2,3,…,k,such that 0<ηj<1 and ∑j=1kηj=1.Furthermore, suppose that the conditionalβ upon Prα=αj, j=1,2,3,…,k, has natural conjugate prior as sj,bj-gamma with p.d.f(20)πβ|α=αj=bjsjΓsjβsj−1exp−bjβ,β>0,bj,sj>0.Combining the likelihood function in (6) and the conditional prior in (20), we get the conditional posterior of β|α=αj as follows:(21)π∗β|α=αj=Lβ,α;xπβ|α=αj∫0∞πβ|α=αjdβ=1Γm+sjvjm+sjβsj−1exp−βvj,where vj=bj+xUmαj.In view of the discrete version of Bayes theorem, we obtain the marginal posterior ofα as(22)hαj|x=pj=Prα=αj|x=Q∫0∞ηjbjsjβm+sj−1αjmujΓsjexp−βvjdβ=QηjbjsjαjmujΓm+sjvjm+sjΓsj,where uj=∏i=1mxUiαj−1 and Q is a normalized constant given by(23)Q−1=∑j=1kηjbjsjαjmujΓm+sjvjm+sjΓsj.


### 4.1. Estimates Based on the Squared-Error Loss Function

From (21) and (22), the Bayes estimates of β,α, and Rt based on the SEL function are derived, respectively, as(24)β^BS=∫0∞∑j=1kpjβπ∗β|α=αj,xdβ,which, upon using (20), simplifies to(25)β^BS=∑j=1kpjsj+mvj,α^BS=∑j=1kpjαj,R^BS=∫0∞∑j=1kpjexp−βtαjπ∗β|α=αj,xdβ=∑j=1kpjvjvj+tαjsj+m.


### 4.2. Estimates Based on the LINEX Error Loss Function

Under the LINEX loss function (12), the Bayes estimate of a function γα,β is given by (13).The Bayes estimator for the scale parameterβ is given by(26)β⌢BL=−1aLn∫0∞∑j=1kpjexp−aβπ∗β|α=αj,xdβ=−1aLn∑j=1kpj1+avj−SJ+m,and the Bayes estimator for α is given by(27)α⌢BL=−1aLn∑j=1kpjexp−aαj.Similarly, the Bayes estimator for the reliability functionRt is given by(28)R⌢tBL=−1aLn∫0∞∑j=1kpjexp−aRtπ∗β|α=αj,xdβ,where Rt is given in (3). By using the exponential series, after some simplifications, we obtain(29)R⌢tBL=−1aLn∑j=1k∑i=0∞−aii!1+itαjvj−m+sj.


### 4.3. Estimates Based on the Weighted LINEX Loss Function

Under the weighted LINEX loss function (14), the Bayes estimate of a function γα,β is given by (18).The Bayes estimator for the scale parameterβ is given by(30)β^WBL=1aLnEexp−zβEexp−βz+a=1aLnI1I2,where(31)I1=∫0∞∑j=1mpjexp−zβπ∗β|α=αj,xdβ=∑j=1mpj1+zvj−SJ+m,I2=∫0∞∑j=1mpjexp−a+zβπ∗β|α=αj,xdβ=∑j=1mpj1+z+avj−SJ+m,and the Bayes estimator for α is given by(32)α^WBL=1aLnEexp−zαEexp−αz+a=1aLnI3I4,where(33)I3=∑j=1kpjexp−zαj,I4=∑j=1kpjexp−a+zαj.Similarly, the Bayes estimatorR^WBLt of the reliability function Rt is given by(34)R^tBL=−1aLn∫0∞∑j=1kpjexp−aRtπ∗β|α=αj,xdβ,R^WBLt=1aLnEexp−zRtEexp−a+zRt=1aLnI5I6,where(35)I5=∑j=1k∑i=0∞−zii!1+itαjvj−m+sj,I6=∑j=1k∑i=0∞−z+aii!1+itαjvj−m+sj.To begin with the calculation, we have to choose the values ofαj,ηj and the hyperparameters sj,bj in the conjugate prior (20), for j = 1, 2, …, k. The values of the previous pairs are quite easy to specify, but to determine sj,bj, there is a need to condition prior ideas about β on each αj, in turn, which is not easy in implementation. An alternative method that is based on the expected value of the reliability function Rt conditional on α=αj to obtain the values sj,bj is given using (20) by(36)Eβ|αjRt|α=αj=bjsjβsj−1Γsjexp−bjβdα=1+tαjbj−sj.So, suppose now that previous ideas about the lifetime distribution enable us to specify two values includingRt1,t1 and Rt2,t2. Therefore, for these prior values Rt=t1 and Rt=t2, the values of sj and bj, for each value of αj, can be computed numerically from (36). In the case of nonexistence of previous ideas, one may use a nonparametric procedure for estimating the corresponding different values of Rt as in Martz and Waller [22].


## 4.1. Estimates Based on the Squared-Error Loss Function

From (21) and (22), the Bayes estimates of β,α, and Rt based on the SEL function are derived, respectively, as(24)β^BS=∫0∞∑j=1kpjβπ∗β|α=αj,xdβ,which, upon using (20), simplifies to(25)β^BS=∑j=1kpjsj+mvj,α^BS=∑j=1kpjαj,R^BS=∫0∞∑j=1kpjexp−βtαjπ∗β|α=αj,xdβ=∑j=1kpjvjvj+tαjsj+m.


## 4.2. Estimates Based on the LINEX Error Loss Function

Under the LINEX loss function (12), the Bayes estimate of a function γα,β is given by (13).The Bayes estimator for the scale parameterβ is given by(26)β⌢BL=−1aLn∫0∞∑j=1kpjexp−aβπ∗β|α=αj,xdβ=−1aLn∑j=1kpj1+avj−SJ+m,and the Bayes estimator for α is given by(27)α⌢BL=−1aLn∑j=1kpjexp−aαj.Similarly, the Bayes estimator for the reliability functionRt is given by(28)R⌢tBL=−1aLn∫0∞∑j=1kpjexp−aRtπ∗β|α=αj,xdβ,where Rt is given in (3). By using the exponential series, after some simplifications, we obtain(29)R⌢tBL=−1aLn∑j=1k∑i=0∞−aii!1+itαjvj−m+sj.


## 4.3. Estimates Based on the Weighted LINEX Loss Function

Under the weighted LINEX loss function (14), the Bayes estimate of a function γα,β is given by (18).The Bayes estimator for the scale parameterβ is given by(30)β^WBL=1aLnEexp−zβEexp−βz+a=1aLnI1I2,where(31)I1=∫0∞∑j=1mpjexp−zβπ∗β|α=αj,xdβ=∑j=1mpj1+zvj−SJ+m,I2=∫0∞∑j=1mpjexp−a+zβπ∗β|α=αj,xdβ=∑j=1mpj1+z+avj−SJ+m,and the Bayes estimator for α is given by(32)α^WBL=1aLnEexp−zαEexp−αz+a=1aLnI3I4,where(33)I3=∑j=1kpjexp−zαj,I4=∑j=1kpjexp−a+zαj.Similarly, the Bayes estimatorR^WBLt of the reliability function Rt is given by(34)R^tBL=−1aLn∫0∞∑j=1kpjexp−aRtπ∗β|α=αj,xdβ,R^WBLt=1aLnEexp−zRtEexp−a+zRt=1aLnI5I6,where(35)I5=∑j=1k∑i=0∞−zii!1+itαjvj−m+sj,I6=∑j=1k∑i=0∞−z+aii!1+itαjvj−m+sj.To begin with the calculation, we have to choose the values ofαj,ηj and the hyperparameters sj,bj in the conjugate prior (20), for j = 1, 2, …, k. The values of the previous pairs are quite easy to specify, but to determine sj,bj, there is a need to condition prior ideas about β on each αj, in turn, which is not easy in implementation. An alternative method that is based on the expected value of the reliability function Rt conditional on α=αj to obtain the values sj,bj is given using (20) by(36)Eβ|αjRt|α=αj=bjsjβsj−1Γsjexp−bjβdα=1+tαjbj−sj.So, suppose now that previous ideas about the lifetime distribution enable us to specify two values includingRt1,t1 and Rt2,t2. Therefore, for these prior values Rt=t1 and Rt=t2, the values of sj and bj, for each value of αj, can be computed numerically from (36). In the case of nonexistence of previous ideas, one may use a nonparametric procedure for estimating the corresponding different values of Rt as in Martz and Waller [22].


## 5. Application with Real Data

To demonstrate the techniques of estimation which are developed in this work, we study the real dataset which was used before by Lawless [23]. These data were taken from the study by Nelson [24]. These data record the time to breakdown of an insulating fluid between electrodes at a voltage of 34 kV (minutes). The 19 times (complete sample) to breakdown are 0.96, 4.15, 0.19, 0.78, 8.01, 31.75, 7.35, 6.50, 8.27, 33.91, 32.52, 3.16, 4.85, 2.78, 4.67, 1.31, 12.06, 36.71, and 72.89.Note: Lawless (1982) applied the ML method on these data to estimate the parametersα, β, and Rt, where β˜=0.145,α˜=0.771, and R˜t=0.606 at t=5 and z=3.Thus, the seven upper record values clearly are 0.96, 4.15, 8.01, 31.75, 33.91, 36.71, and 72.89.A Weibull distribution represents the time to breakdown for a fixed voltage level as it was mentioned by engineering considerations. The calculations for this application are conducted by using the gamma prior for the scale parameter, while the discrete distribution is applied for the shape parameter. The hyperparameters of the gamma prior (36) and the values of αj are derived by the following steps:(1)
Depending on the previous seven upper records, we use a nonparametric procedureRti=XUi=m−i+0.625/m+0.25,i=1,2,3,…,m, with m=7 to estimate two values of the reliability function. Therefore, the reliability function for times t1=8.01andt2=33.91 is Rt1=0.638 and Rt2=0.362, respectively.(2)
Assuming thatαj 0.5, 0.55, 0.6, … , 0.95, the value of the MLE of parameter α from (8) is computed α^Ml=0.599.(3)
The two prior values obtained in Step 1 are substituted into (34), where sj and bj are solved numerically for each given αj,j=1,2,3,…,10, using the Newton–Raphson method.Table1 shows the values of the hyperparameters and the posterior probabilities derived for each value in αj, while Table 2 displays the computed estimations of α, β, and Rt using ML estimates .ML and the Bayes estimates (.BS, .BL, and .WBL).Table 1
Prior information, hyperparameters, and posterior probabilities.
jαjηjsjbjujpj10.50.13006041.66018∗1060.0001040.00508720.550.11.33736∗1069.01977∗1060.0002610.00598430.600.15.7326642.72120.0006530.03415740.650.12.4259418.99450.0016320.09753050.700.11.5669812.91430.0040830.14918160.750.11.1708810.17310.0102130.17126670.800.10.9422728.64220.0255450.16761680.850.10.7930597.686990.0638970.14855190.900.10.6877477.051940.1598280.123184100.950.10.6092726.614380.3997850.097443Table 2
Estimates ofα, β, and Rt when t=5 and z=3.
Methodα^α^−α˜β^β^−β˜R^tR^t−R˜t.ML0.5990.1720.5360.3910.2450.361.BS0.7860.0150.2020.0570.5140.092.BLa=−10.7910.0200.2060.0610.5210.085.BLa=−20.7960.0250.2100.0650.5290.077.BLa=20.7760.0050.1950.0500.4980.108.WBLa=−10.7610.0100.1850.0400.4750.131.WBLa=−20.7660.0050.1880.0430.4830.123.WBLa=20.7460.0250.1760.0310.4520.154According to the smallest differencesβ^−β˜ and α^−α˜, the developed method in this work (WLINEX) has the best performance in estimating the scale parameter β and shape parameter α as it is shown in Table 2. With respect to the reliability function Rt, the developed method is better than the ML method and comes in the third level of accuracy after BL and BS methods with small differences. Finally, these results reveal that the proposed model in this work is promising and can be applied in other real environments.



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