Abstract
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. This encouraged us to look for mathematical method that gives and guarantees the best values of the weighted coefficients. The 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. The evaluation was done based on absolute bias and mean square errors. The 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.
Highlights
Scholars always concern about how to find the best estimates of parameters and reliability function of the probability distributions
Nonlinear programming was employed to get the best values of weighted coefficients (ω1 and ω2) 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: (1) All tables showed that the Bayes estimates under balanced linear exponential (BLINEX) loss function are the best according to the smallest values of absolute bias and mean square error (MSE) comparing with the estimates under linear exponential (LINEX) loss function, balanced square error (BSE) loss function, Squared Error (SE) loss function, or maximum likelihood estimate (MLE)
Summary
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. Many authors prefer using a squared error loss function to produce Bayesian estimates This loss function has mainly criticized where both of overestimation and underestimation are given equal importance, which does not agree with real practices. 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.
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