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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.

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. There 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 [

After that, the balanced loss function idea appeared in the literature which tried to reflect the desired criteria of two methods (see equation (

However, the majority of proposed balanced loss functions in the literature determine the value of weighted coefficients

The 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 [

The probability density function (pdf) and cumulative distribution function (cdf) of the IRD with scale parameter

Moreover, the reliability function

Let

Assuming that

The log-likelihood function is written as

By differentiating (

By using the invariance property of the maximum likelihood estimator, the maximum likelihood estimator of reliability function

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.

SE loss function is a symmetric loss function. The SE loss function is expressed as follows:

LINEX loss function is an asymmetric loss function. The LINEX loss function is expressed as follows (see Varian [

BLF is a mix of two estimators. In general, BLF is expressed as follows (see Jozani et al. [

BSE loss function is obtained by choosing

Note that when

The BLINEX loss function is obtained by choosing

And the corresponding Bayes estimate of the unknown parameter

It is worth noting, when

In this section, we derive the Bayes estimates of the scale parameter

By combining the likelihood function in equation (

Based on BSE loss function and by using equation (

Based on the BSE loss function and by using equation (

Similarly, the Bayes estimate

In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient

Based on BLINEX loss function and by using equation (

Based on BLINEX loss function and by using equation (

Similarly, the Bayes estimate

In this work, we solve the following nonlinear programming (using Mathematica software) to find the optimal values of the weighted coefficient

All estimation methods, mentioned in Section

For the given values of prior parameters

Using

The different estimates of

Steps 1 to 3 are repeated 10,000 times.

The evaluation is done depending on the absolute bias in addition to the mean square error (MSE)

where

The results are listed in Tables

Absolute bias of the estimates of

3 | 1.03787 | 0.60693 | 0.56808 | 0.60662 | 0.39863 | 0.31514 | 0.56787 | 0.39849 | 0.28874 |

4 | 0.78828 | 0.55394 | 0.51968 | 0.55372 | 0.39272 | 0.32124 | 0.51954 | 0.39257 | 0.29764 |

5 | 0.65342 | 0.50596 | 0.47534 | 0.50578 | 0.37810 | 0.31704 | 0.47525 | 0.37793 | 0.29541 |

6 | 0.55881 | 0.46009 | 0.43195 | 0.45996 | 0.35661 | 0.30454 | 0.43188 | 0.35644 | 0.28420 |

7 | 0.48079 | 0.41094 | 0.38500 | 0.41083 | 0.32850 | 0.28677 | 0.38495 | 0.32832 | 0.26636 |

MSEs of the estimates of

3 | 4.55002 | 0.66780 | 0.65187 | 0.66706 | 0.26109 | 0.14830 | 0.65120 | 0.26109 | 0.14235 |

4 | 1.82889 | 0.58323 | 0.58270 | 0.58270 | 0.26526 | 0.15928 | 0.56883 | 0.26526 | 0.15391 |

5 | 1.13061 | 0.49605 | 0.48381 | 0.49566 | 0.25113 | 0.15827 | 0.48348 | 0.25113 | 0.15330 |

6 | 0.77355 | 0.41691 | 0.40581 | 0.41662 | 0.22785 | 0.14891 | 0.40557 | 0.22785 | 0.14420 |

7 | 0.54374 | 0.34428 | 0.33437 | 0.34407 | 0.19897 | 0.13374 | 0.33420 | 0.19897 | 0.12880 |

Absolute bias of the estimates of

3 | 0.05343 | 0.03276 | 0.03090 | 0.03444 | 0.03276 | 0.03120 | 0.03195 | 0.03090 | 0.02985 |

4 | 0.04251 | 0.03013 | 0.02848 | 0.03137 | 0.03013 | 0.02896 | 0.02918 | 0.02848 | 0.02777 |

5 | 0.03582 | 0.02769 | 0.02622 | 0.02866 | 0.02769 | 0.02677 | 0.02670 | 0.02621 | 0.02571 |

6 | 0.03093 | 0.02529 | 0.02394 | 0.03444 | 0.03276 | 0.03120 | 0.03195 | 0.03090 | 0.02985 |

7 | 0.02678 | 0.02267 | 0.02142 | 0.03137 | 0.03013 | 0.02896 | 0.02918 | 0.02848 | 0.02777 |

MSEs of the estimates of

3 | 0.00815 | 0.00188 | 0.00184 | 0.00209 | 0.00188 | 0.00209 | 0.00203 | 0.00184 | 0.00167 |

4 | 0.00457 | 0.00166 | 0.00163 | 0.00181 | 0.00166 | 0.00153 | 0.00176 | 0.00163 | 0.00151 |

5 | 0.00304 | 0.00143 | 0.00140 | 0.00154 | 0.00143 | 0.00133 | 0.00149 | 0.00140 | 0.00131 |

6 | 0.00216 | 0.00121 | 0.00119 | 0.00130 | 0.00121 | 0.00114 | 0.00126 | 0.00119 | 0.00112 |

7 | 0.00158 | 0.00101 | 0.00099 | 0.00107 | 0.00101 | 0.00095 | 0.00104 | 0.00099 | 0.00094 |

In this paper, nonlinear programming was employed to get the best values of weighted coefficients (

The results are listed in Tables

All tables showed that the Bayes estimates under BLINEX loss function are the best according to the smallest values of absolute bias and MSE comparing with the estimates under LINEX loss function, BSE loss function, SE loss function, or MLEs. Bayes estimates under the BSE loss function came in the second level of accuracy. The third, fourth, and fifth levels of accuracy were for Bayes estimates under the LINEX loss function, the estimates under SE loss function, and ML estimations, respectively.

The results showed that the values of all MSEs and all absolute bias decrease as

In order to show the effect of the shape parameter of the asymmetric loss function, we examined different values of

The data were generated by simulation done by using mathematical software.

The authors declare that they have no conflicts of interest.