Nonlinear Programming to Determine Best Weighted Coefficient of Balanced LINEX Loss Function Based on Lower Record Values

Majority research studies in the literature determine the weighted coefficients of balanced loss function by suggesting some arbitrary values and then conducting comparison study to choose the best. However, this methodology is not efficient because there is no guarantee ensures that one of the chosen values is the best. )is encouraged us to look for mathematical method that gives and guarantees the best values of the weighted coefficients. )e proposed methodology in this research is to employ the nonlinear programming in determining the weighted coefficients of balanced loss function instead of the unguaranteed old methods. In this research, we consider two balanced loss functions including balanced square error (BSE) loss function and balanced linear exponential (BLINEX) loss function to estimate the parameter and reliability function of inverse Rayleigh distribution (IRD) based on lower record values. Comparisons are made between Bayesian estimators (SE, BSE, LINEX, and BLINEX) and maximum likelihood estimator via Monte Carlo simulation. )e evaluation was done based on absolute bias and mean square errors. )e outputs of the simulation showed that the balanced linear exponential (BLINEX) loss function has the best performance. Moreover, the simulation verified that the balanced loss functions are always better than corresponding loss function.


Introduction
Scholars always concern about how to find the best estimates of parameters and reliability function of the probability distributions. For this purpose, many methods were proposed. Some of these methods are classical, where they depend only on the sample information under study, assuming that the distribution parameter is fixed but unknown. ere are other approaches (which are commonly known as Bayesian methods) that depend on merging prior information with sample information, assuming that the prior parameters behave as random variables, which are commonly known as prior probability distributions.
From a Bayesian perspective, the choice of the loss function is a crucial part of the estimation and prediction problems. To simplify the calculations, many authors prefer using a squared error loss function to produce Bayesian estimates. However, this loss function has mainly criticized where both of overestimation and underestimation are given equal importance, which does not agree with real practices. To deal with this situation, several asymmetric loss functions were proposed in the literature. For example, general entropy loss function (Abdel-Hamid [1]) and linear exponential (LINEX) loss function (Al-Duais and Alhagyan [2]; Khatun and Matin [3]) After that, the balanced loss function idea appeared in the literature which tried to reflect the desired criteria of two methods (see equation (13)), for example, balanced square error (BSE) loss function [4], balanced general entropy (BGE) loss function [5]), and balanced linear exponential (BLINEX) loss function [6] EL-Sagheer [7] EL-Sagheer [8].
However, the majority of proposed balanced loss functions in the literature determine the value of weighted coefficients ω 1 and ω 2 randomly without convinced mathematical justification. is motivated us to treat this issue by determining the weighted coefficients by using nonlinear programming. In this paper, we are going to use two balanced loss functions (i.e., BSE and BLINEX) to estimate the parameter and reliability function of inverse Rayleigh distribution (IRD) based on lower record values utilizing nonlinear programming in determining the best-weighted coefficients.
e IRD is considered as one of the important distributions. It has wide applications in the area of reliability theory, survival analysis, and life testing study. IRD under record value was studied by Muhammad [9]; Shawky and Badr [10]; Soliman et al. [11]; Manzoor et al. [12]; Rasheed and Aref [13]; and Abdullah and Aref [14]. e probability density function (pdf ) and cumulative distribution function (cdf ) of the IRD with scale parameter α are given, respectively, as follows: Moreover, the reliability function R(t) at mission time t for the IRD is given by

Record Values and Maximum Likelihood (ML) Estimation
Let X 1 , X 2 , X 3 , . . . . . . be a sequence of independent and identically distributed (iid) random variables with (cdf ) F(x) We say that X j is a lower record and denoted by Assuming that X L(1) , X L(2) , X L(3) , . . . X L(m) are the first m lower record values arising from a sequence X i of iid inverse Rayleigh distribution whose pdf and cdf are, respectively, given by (1) where f(.) and F(.) are given, respectively, by (1) and (2) after replacing x by x L(i) . e likelihood function based on the m lower record values x is given by e log-likelihood function is written as By differentiating (6) with respect to the parameterα and equating to zero, the maximum likelihood estimate (MLE), under lower record value, say α ML , was obtained as By using the invariance property of the maximum likelihood estimator, the maximum likelihood estimator of reliability function R(t) ML of R(t) given by (3) after replacing α by α ML is

Loss Functions
From a Bayesian perspective, the choice of loss function is an essential part in the estimation and prediction problems. In this work, we use three main types of loss function including squared error loss function, LINEX loss function, and balanced loss functions.

Squared Error (SE) Loss
Function. SE loss function is a symmetric loss function. e SE loss function is expressed as follows: where ϕ is the estimation of parameter ϕ. e Bayes estimator of ϕbased on SE loss function denoted byϕ SE is obtained as follows: 2 Complexity

Linear Exponential (LINEX) Loss Function. LINEX loss
function is an asymmetric loss function. e LINEX loss function is expressed as follows (see Varian [15]): where Δ � (ϕ − ϕ). e sign and magnitude of c reflect the direction and degree of asymmetry, respectively. e Bayes estimator related to LINEX loss function, denoted by ϕ L , is given by provided that E ϕ (exp − cϕ ) exists and finite, where E ϕ denotes the expected value.

Balanced Loss Function (BLF)
. BLF is a mix of two estimators. In general, BLF is expressed as follows (see Jozani et al. [16]): where ρ is an arbitrary loss function, while c 0 is a chosen a prior target estimator of ϕ that can be obtained by several methods like maximum likelihood, least squares, or unbiasedness, and ω 1 and ω 2 represent weighted coefficient ω 1 and ω 2 ϵ [0, 1). In this work, we focus on two types of BLF, including balanced squared error (BSE) loss function and balanced LINEX (BLINEX) loss function.

Balanced Squared Error (BSE) Loss
Function. BSE loss function is obtained by choosingρ(ϕ , c) � (c − ϕ) 2 , so equation (13) will be on the form (see Ahmadi et al. [17]): and the corresponding Bayes estimate of the unknown parameter ϕ is given by Note that when ω 1 � 0, then BSE loss function is just an SE loss function.

Balanced Linear Exponential (BLINEX) Loss Function.
e BLINEX loss function is obtained by choosing ρ(ϕ , c) (13) as follows (see Zellner [18]): And the corresponding Bayes estimate of the unknown parameter ϕ is given by It is worth noting, when ω 1 � 0 then BLINEX loss function is just a LINEX loss function.

Bayes Estimation
In this section, we derive the Bayes estimates of the scale parameter α and the reliabilityR(t) function of the IRD by using balanced loss functions (BLF). Furthermore, we assume gamma (8, b) as a conjugate prior distribution for α as follows: By combining the likelihood function in equation (5) with the prior pdf of α in equation (18), we get the posterior distribution of α as where

Estimates Based on Balanced Squared Error (BSE) Loss
Function. Based on BSE loss function and by using equation (15), the Bayes estimate of a function ϕ where ϕ can be α or R(t) is given by where ϕ ML is the ML estimate ofϕ and E(ϕ| x) can be obtained by Based on the BSE loss function and by using equation (21), the Bayes estimator α BSE for α is where α ML is the ML estimate ofα, which can be obtained using equation (7) and E(α| x) can be obtained using the following equation: Similarly, the Bayes estimate R(t) BSE of the reliability R(t) at a mission time t related to BSE loss function is Complexity where R(t) ML is the ML estimate of R(t) which can be obtained using equation (8) and E(R(t)| x) can be obtained using the following equation: In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient ω 1 and ω 2 in equation (21):

Estimates Based on Balanced Linear Exponential (BLI-NEX) Loss Function.
Based on BLINEX loss function and by using equation (16), the Bayes estimate of a function ϕ where ϕ can be α or R(t) is given by where ϕ ML is the ML estimate of ϕ and E(exp[− c ϕ]| x) can be obtained by Based on BLINEX loss function and by using equation (28), the Bayes estimator α BL for α is given as where α ML is the ML estimate of α which can be obtained using equation (7) and E(exp[− c α]| x) can be obtained using the following integral: Similarly, the Bayes estimate R(t) BL of the reliability R(t) at a mission time t related to BLINEX loss function is where R(t) ML is the ML estimate of R(t) which can be obtained using equation (8) and E(exp[− c R(t)]| x)can be obtained using the following integral: In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient ω 1 and ω 2 in equation (28):

Simulation Study and Comparisons
All estimation methods, mentioned in Section 4, are used to estimate the parameter and reliability function of IRD. To examine the performance of these estimation methods, the Monte Carlo simulation study is conducted. e simulation consists of four steps as follows: (1) For the given values of prior parameters ( b � 2 , 8 � 1), generate a random value α � 1.383 from the Gamma prior pdf in equation (18) [19].

Complexity
(2) Using α obtained in Step 1, we generatem (m � 3, 4, 5, 6, 7) lower record values from inverse Rayleigh distribution whose pdf is given by equation where ϕ is the estimate at the i th run.

Concluding Remarks
In this paper, nonlinear programming was employed to get the best values of weighted coefficients (ω 1 and ω 2 ) of the balanced loss function. e Bayesian and non-Bayesian estimates of the parameter α and reliability function R(t) of the lifetimes follow the inverse Rayleigh distribution. e estimations were conducted depending on lower record values.
e results are listed in Tables 1-4. e main observations are stated in the following points: ( Data Availability e data were generated by simulation done by using mathematical software.

Conflicts of Interest
e authors declare that they have no conflicts of interest.