Dynamic Analysis, Learning, and Robust Control of Complex SystemsView this Special Issue
Bayesian Estimations under the Weighted LINEX Loss Function Based on Upper Record Values
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.
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 , inverse Weibull [10, 11], exponentiated Family [12, 13], Lomax [14, 15], power Lindley model , exponentiated Weibull [17, 18], and normal distribution .
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.
Let be a sequence of independent and identically distributed (iid) random variables with (c.d.f) and (p.d.f) . Set , , and we say that is an upper record and denoted by if .
Assume that are the first -upper record values arising from a sequence of (iid) Weibull variables with probability density function (p.d.f)and cumulative distribution function (c.d.f)where and 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
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 first -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 instead of . We have the joint density function of the first m-upper record values which is given according to Arnold et al.  bywhere , and are given, by (1) and (2), respectively, after replacing by .
The likelihood function based on the -upper record values is given as follows:
When both and are unknown, the maximum likelihood estimates of and can be obtained from (7) by solving the following two equations:
By the invariance property of the MLE, reliability function of can be obtained as given by (3) after replacing and by:
3. Loss Function
We consider three different loss functions.
3.1. Squared-Error Loss Function (SEL)
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 estimate under the above loss function is the posterior mean, i.e., .
3.2. LINEX Loss Function
The LINEX loss function for can be expressed as where . Here, represents the shape parameter of the loss function. The behavior of the LINEX loss function changes with the choice of . In particular, if is close to zero, this loss function is close to the SEL loss function and therefore almost symmetric.
The Bayes estimator of denoted by, depending on the LINEX loss function, is given byprovided that exists and is finite, where 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:where represents the estimated parameter that makes the expectation of the loss function (equation (14)) as small as possible. The value represents the proposed weighted function, which is given as follows:
Depending on the posterior distribution of parameter and by using the proposed weighted function, we get the Bayesian estimation of parameter as follows:
So, we can find that
Consequently, the Bayesian estimation of parameter using WLINEX will be
4. Bayes Estimation
In the case of the two-parameter problem, we need to specify a general joint prior for and which may lead to complicated calculations. Aiming to solve this problem, Soland’s method was used. Soland  considered a family of joint prior distributions that places continuous distributions on the scale parameter and discrete distributions on the shape parameter.
Suppose that is restricted to the values with prior probabilities , that is,such that and .
Furthermore, suppose that the conditional upon , , has natural conjugate prior as -gamma with p.d.f
In view of the discrete version of Bayes theorem, we obtain the marginal posterior of aswhere and Q is a normalized constant given by
4.1. Estimates Based on the Squared-Error Loss Function
4.2. Estimates Based on the LINEX Error Loss Function
The Bayes estimator for the scale parameter is given byand the Bayes estimator for is given by
Similarly, the Bayes estimator for the reliability function is given bywhere is given in (3). By using the exponential series, after some simplifications, we obtain
4.3. Estimates Based on the Weighted LINEX Loss Function
The Bayes estimator for the scale parameter is given bywhereand the Bayes estimator for is given bywhere
Similarly, the Bayes estimator of the reliability function is given bywhere
To begin with the calculation, we have to choose the values of and the hyperparameters in the conjugate prior (20), for j = 1, 2, …, k. The values of the previous pairs are quite easy to specify, but to determine , there is a need to condition prior ideas about on each , in turn, which is not easy in implementation. An alternative method that is based on the expected value of the reliability function conditional on to obtain the values is given using (20) by
So, suppose now that previous ideas about the lifetime distribution enable us to specify two values including and . Therefore, for these prior values and , the values of and , for each value of , 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 as in Martz and Waller .
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 . These data were taken from the study by Nelson . 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 parameters , , and , where , and at and .
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 are derived by the following steps:(1)Depending on the previous seven upper records, we use a nonparametric procedure , with to estimate two values of the reliability function. Therefore, the reliability function for times is and , respectively.(2)Assuming that 0.5, 0.55, 0.6, … , 0.95, the value of the MLE of parameter from (8) is computed .(3)The two prior values obtained in Step 1 are substituted into (34), where and are solved numerically for each given , using the Newton–Raphson method.
Table 1 shows the values of the hyperparameters and the posterior probabilities derived for each value in , while Table 2 displays the computed estimations of , , and using ML estimates and the Bayes estimates (, , and ).
According to the smallest differences and , the developed method in this work (WLINEX) has the best performance in estimating the scale parameter and shape parameter as it is shown in Table 2. With respect to the reliability function , 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.
The data were taken from the study by Nelson . These data record the time to breakdown of an insulating fluid between electrodes at a voltage of 34 kV (minutes).
Conflicts of Interest
The author declares that there are no conflicts of interest.
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