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

Al-Duais, Fuad S.

doi:https://doi.org/10.1155/2021/6648462

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments Supplementary Materials References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6648462 | https://doi.org/10.1155/2021/6648462

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

Fuad S. Al-Duais¹

Academic Editor: Akif Akgul

Received03 Oct 2020

Revised12 Jan 2021

Accepted09 Mar 2021

Published25 Mar 2021

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.

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:

The reliability function is given as follows:

Here, and 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 ( and ) 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

Let be a sequence of independent and identically distributed random variables with CDF and PDF . Set , , and say that is a lower record and denoted by if . Suppose we observe the first lower record values 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 lower record values as given by Sultan [24]:

Here, and are given by (1) and (2), respectively, after replacing by . The likelihood function based on the lower record values x is given as follows:

We obtain that the log-likelihood function may be written as follows:

Assuming that the shape parameter is known, using equation (6), the maximum likelihood estimator (MLE) of the scale parameter can be shown to be of the following form:

If is replaced by in equation (3), we can obtain the MLE of reliability function of R (t) depending on the invariance property:

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

The Bayes estimator of based on SE loss function can be obtained as follows:

2.2.2. Linear Exponential Loss Function

Varian [25] introduced the LINEX loss function. LINEX is an asymmetric loss function that can be expressed aswhere . The sign and magnitude of reflect the direction and degree of asymmetry, respectively. The Bayes estimator relative to LINEX loss function, denoted by , is given as follows:provided that exists and is finite, where 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:

Here, represents the estimated parameter that makes the expectation of loss function by equation (13) as small as possible. The value represents the proposed weighted function, which is equal to the following:

Depending on the posterior distribution of the parameter and using the proposed weighted function as in equation (14), we can attain the estimated weighted Bayes of the parameter as follows:

It is known that, to find the value of that minimizes , we have to perform the following two steps: (i) Therefore, we can find the following: Consequently, the Bayesian estimation of the parameter using WLINEX will be(i) and exist and are finite, where denotes the expected value.(ii) at the minimum value computed by (i):

Because satisfies conditions (i) and (ii), it follows that 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 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 formwhere represents an estimator of parameter is a chosen prior estimator of that can be obtained by several methods such as maximum likelihood (ML) or least squares, and represent weighted coefficients belonging to , is an arbitrary loss function when is estimated by , and 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.1Balanced Squared Error Loss Function. BSE loss function is obtained by choosing and in equation (20) as follows: The Bayes estimation of under is given by Note that SE loss function is a special case of BSE loss function when .2Balanced Linear Exponential Loss Function. The BLINEX is obtained by choosing and in equation (20) as follows: The Bayes estimation under is given by Note that when , then BLINEX is exactly LINEX loss function.3Balanced Weighted Linear Exponential (BWLINEX) Loss Function. The BWLINEX is obtained by choosingin equation (20) as follows:and the Bayes estimation under is given by

Note that when , the BWLINEX loss function is exactly WLINEX loss function.

2.3. Bayes Estimation

In this section, we derive Bayes estimates of the scale parameter and the reliability of the IWD. We use six different loss functions, including SE, LINEX, WLINEX, BSE, BLINEX, and BWLINEX. Under the assumption that the shape parameter is known, we assume a gamma (conjugate prior) for density for with parameters and :

Combining the likelihood function in equation (5) with the prior PDFof in equation (28), we get the posterior of aswhere ,

2.3.1. Estimates Based on Balanced Squared Error Loss Function

Based on BSE and using equation (21), the Bayes estimation of a parameter (which can be the scale parameter or the reliability function ) is given bywhere is the ML estimate of and can be obtained using

Note 1. When in equation (31), the Bayes estimation under BSE loss function of and denoted by is given bywhere is the ML estimate of , which can be obtained using equation (7). can be obtained usingand is given by equation (28).

Note 2. When in equation (29), the Bayes estimation under BSE loss function of and denoted by is given bywhere is the ML estimate of and can be obtained using equation (8), and can be obtained using the following:

The main contribution of this work is to use nonlinear programming to find the optimal values of and to compute and in equations (33) and (35), respectively. To achieve this target, we minimize the MSE as follows:

2.3.2. Estimates Based on BLINEX Loss Function

Based on BLINEX and using equation (24), the Bayes estimation of a parameter (which can be the scale parameter or the reliability function ) is given bywhere is the ML estimate, and can be obtained using

Note 1. When in equation (38), the Bayes estimation under BINEX loss function of and denoted by is given bywhere is the ML estimate of and can be obtained using equation (7), and can be obtained using

Note 2. When in equation (38), then the Bayes estimation under BINEX loss function of , which is denoted by , is given bywhere is the ML estimate of and can be obtained using equation (8), and can be obtained using

Again, we use nonlinear programming to find the optimal values of and to compute and in equations (41) and (43), respectively. To achieve this target, we minimize the MSE as follows:

2.3.3. Estimates Based on Weighted Balanced Loss Function

Based on WBLINEX and using equation (26), the Bayes estimation of a parameter (which can be the scale parameter or the reliability function ) is given bywhere is the ML estimate of and and can be obtained by

Note 1. When in equation (45), the Bayes estimation under WBINEX loss function of and denoted by is given bywhere is the ML estimate of and can be obtained using equation (7) and can be obtained as follows, respectively:

Note 2. When in equation (45), then the Bayes estimation under WBINEX loss function of , which is denoted by , is given bywhere is the ML estimate of , which can be obtained using equation (8), and can be obtained as follows, respectively:

As in equations (37) and (44), we minimize the MSE to compute and in equations (47) and (49) as follows, respectively:

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, including and . We selected the values of () as and . We selected the positive and negative values to represent both cases of upper estimate and lower estimate, respectively, while the chosen values of are 3 and −3.

We conducted the simulation according to the following steps:(1)For the given values , we generated a random value from the prior PDF as in equation (27).(2)By using the value from Step 1 with selected values of , we generated data of lower record values from IWD (equation (1) for different sizes and 7).(3)We computed the estimates of at a chosen time of using the estimations under the study.(4)We repeated Steps 1–3 10,000 times and calculated the MSE for each estimate (say ) usingwhere can be or and is the estimate at the run.

3. Results and Discussion

The results of the simulation are listed in Tables 1–4.

In this paper, we employed nonlinear programming to obtain the best values of weighted coefficients ( and ) of the balanced loss function. The estimates of the parameter 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)Tables 1–4 reveal that the proposed estimator BWLINEX is the absolute best method to estimate scale parameter and reliability according to the smallest values of MSEs compared with the estimators of all other methods under study.(2)For unbalanced loss functions, Tables 1 and 3 showed that the proposed estimator WLINEX is the best method to estimate scale parameter and reliability 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 as increases. This means that there is an inverse relationship between the evaluation functions and the number of recorded values.(4)It is clear from Table 1 that the ML method has the worst performance in estimating the scale parameter , 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)Tables 1 and 2 show that the lower estimate () of the scale parameter λ is better than the upper estimate ( and ) 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 parameter .(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.

Data Availability

The data were generated by simulation done by mathematical software. The simulation is in the Supplementary materials.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This project was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University under the research project no. 2020/01/12030.

Supplementary Materials

The contribution of this paper depends on two parts including theoretical part and empirical part. The data in empirical part were generated by Monte Carlo simulation. (Supplementary Materials)

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Copyright

Copyright © 2021 Fuad S. Al-Duais. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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