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

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.

IWD is one of the most widely used probability distributions with many real environment applications. is 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. e probability density function (PDF) and cumulative distribution function (CDF) of IWD are given as follows, respectively: e reliability function is given as follows: 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. e main criticism of this approach is that SE gives overestimation and underestimation equal importance.
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. ese include balanced squared error (BSE) loss function and balanced linear exponential (BLINEX) loss function [21][22][23].

Methodology
e weighted coefficients (ω 1 and ω 2 ) in balanced loss functions are routinely determined by arbitrary choice without depending upon any mathematical justification.
is motivated us to propose a justifiable mathematical approach to determine these coefficients. e 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).

Record Values and Maximum Likelihood Estimation.
Let X 1 , X 2 , X 3 , . . . . . . be a sequence of independent and identically distributed random variables with CDF F(x) and PDF f(x). Set Y n � min(X 1 , X 2 , X 3 , . . . , X n ), n ≥ 1, and say that X j is a lower record and denoted by X L(i) if Y j < Y j− 1 , j > 1. Suppose we observe the first n lower record values X L(1) , X L(2) , X L(3) , . . . , X L(n) 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 ≡ (x L(1) , x L(2) , x L(3) , . . . . . . , x L(n) ) as given by Sultan [24]: Here, f(.) and F(.) are given by (1) and (2), respectively, after replacing x by x L(i) . e likelihood function based on the n lower record values x is given as follows: We obtain that the log-likelihood function may be written as follows: ln x L(i) . (6) Assuming that the shape parameter θ 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: If λ is replaced by λ ML in equation (3), we can obtain the MLE of reliability function R ML (t) of R (t) depending on the invariance property:

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

Squared Error Loss Function.
e SE loss function can be expressed as e Bayes estimator of φ based on SE loss function can be obtained as follows:

Linear Exponential Loss Function.
Varian [25] introduced the LINEX loss function. LINEX is an asymmetric loss function that can be expressed as where Δ � (φ − φ). e sign and magnitude of c reflect the direction and degree of asymmetry, respectively. e Bayes estimator relative to LINEX loss function, denoted by φ LINEX , is given as follows: provided that E φ � (e − cφ ) exists and is finite, where E φ denotes the expected value.

Weighted Linear Exponential Loss
Functions. e researcher proposes this loss function depending on WLI-NEX loss function as follows: Here, φ represents the estimated parameter that makes the expectation of loss function by equation (13) as small as possible. e value w(φ) represents the proposed weighted function, which is equal to the following: Depending on the posterior distribution of the parameter φ and using the proposed weighted function as in equation (14), we can attain the estimated weighted Bayes of the parameter φ as follows: It is known that, to find the value of φ that minimizes EL ⋆ w (φ, φ), we have to perform the following two steps: erefore, we can find the following: Consequently, the Bayesian estimation of the parameter φ using WLINEX will be > 0 at the minimum value computed by (i): Because φ 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).

Balanced Loss
Function. According to AbdEllah [22], the class of balanced loss function (BLF) can be written in the form where τ represents an estimator of parameter ξ(φ), τ 0 is a chosen prior estimator of ξ(φ) that can be obtained by several methods such as maximum likelihood (ML) or least squares, ω 1 and ω 2 represent weighted coefficients belonging to [0, 1), ρ(ξτ 0 , τ) is an arbitrary loss function when ξ(φ) is estimated by τ, and k(φ) 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.

Bayes Estimation.
In this section, we derive Bayes estimates of the scale parameter λ and the reliability R(t) 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 ] and η: Combining the likelihood function in equation (5) with the prior PDFof λ in equation (28), we get the posterior of λ as where X � ( x L(1) , x L(2) , x L(3) , . . . . . . , x L(n) ),

Estimates Based on Balanced Squared Error Loss
Function. Based on BSE and using equation (21), the Bayes estimation of a parameter c (which can be the scale parameter λ or the reliability function R(t)) is given by where c ML is the ML estimate of c and E(c|X) can be obtained using Note 1. When c � λ in equation (31), the Bayes estimation under BSE loss function of λ and denoted by λ BSE is given by where λ ML is the ML estimate of λ, which can be obtained using equation (7). E(λ|X) can be obtained using and D is given by equation (28).

Note 2.
When c � R(t) in equation (29), the Bayes estimation under BSE loss function of R(t) and denoted by R BSE (t) is given by where R ML (t) is the ML estimate of R(t) and can be obtained using equation (8), and E(R(t)|X) can be obtained using the following: e main contribution of this work is to use nonlinear programming to find the optimal values of ω 1 and ω 2 to compute λ BSE and R BSE (t) in equations (33) and (35), respectively. To achieve this target, we minimize the MSE as follows: (24), the Bayes estimation of a parameter c (which can be the scale parameter λ or the reliability function R(t)) is given by

Estimates Based on BLINEX Loss Function. Based on BLINEX and using equation
where c ML is the ML estimate, and E(exp[− cc]|X) can be obtained using Note 1. When c � λ in equation (38), the Bayes estimation under BINEX loss function of λ and denoted by c BLINEX is given by where λ ML is the ML estimate of λ and can be obtained using equation (7), and E(exp[− cλ]|X) can be obtained using Note 2. When c � R(t) in equation (38), then the Bayes estimation under BINEX loss function of R(t), which is denoted by R BLINEX (t), is given by where R ML (t) is the ML estimate of R(t) and can be obtained using equation (8), and E(exp Again, we use nonlinear programming to find the optimal values of ω 1 and ω 2 to compute λ BLINEX and R BLINEX (t) in equations (41) and (43), respectively. To achieve this target, we minimize the MSE as follows: (44)

Estimates Based on Weighted Balanced Loss Function.
Based on WBLINEX and using equation (26), the Bayes estimation of a parameter c (which can be the scale parameter λ or the reliability function R(t)) is given by where c ML is the ML estimate of c and I 1 and I 2 can be obtained by Note 1. When c � λ in equation (45), the Bayes estimation under WBINEX loss function of λ and denoted by λ WBLINEX is given by where λ ML is the ML estimate of λ and can be obtained using equation (7) and I 3 and I 4 can be obtained as follows, respectively: (49)

Mathematical Problems in Engineering
Note 2. When c � R(t) in equation (45), then the Bayes estimation under WBINEX loss function of R(t), which is denoted by R BLINEX (t), is given by where R ML (t) is the ML estimate of R(t), which can be obtained using equation (8), and I 5 and I 6 can be obtained as follows, respectively: As in equations (37) and (44), we minimize the MSE to compute λ WBLINEX and R WBLINEX (t) in equations (47) and (49) as follows, respectively:

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 c 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 values (η � 2, ] � 1), we generated a random value λ � 1.383 from the prior PDF as in equation (27).
(3) We computed the estimates of λ, R(t) 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 (say φ) using where φ can be λ or R(t) and φ i is the estimate at the i th run.

Results and Discussion
e 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 (ω 1 and ω 2 ) of the balanced loss function. e estimates of the parameter λ and reliability function R (t) follow the IWD. e estimation methods under study include ML, SE, LINEX, WLINEX, BSE, BINEX, and BWLINEX. We conducted the estimations depending on lower record values. e main observations of the results are stated in the following points: (

Conclusion
In this work, we developed LINEX to estimate the scale parameter and reliability function of IWD depending on lower record values. e development occurred through merging a weight into LINEX to produce a new loss function called WLINEX. We used WLINEX to derive the scale Table 4: MSEs of the estimates of R(t) under balanced loss functions at z � − 3.    parameter and reliability function of the IWD. e 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: WLI-NEX and BWLINEX. We then compared the proposed methods with the other methods, including ML, SE, LINEX, BSE, and BLINEX. e 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. is result shows that the approach followed is promising and can be applied in a real environment.
Data Availability e data were generated by simulation done by mathematical software. e simulation is in the Supplementary materials.

Conflicts of Interest
e author declares that he has no conflicts of interest.