Comparison of classical and Bayesian estimators to estimate the parameters in Weibull distribution under weighted general entropy loss function

In this work, we have developed a General Entropy loss function (GE) to estimate parameters of Weibull distribution (WD) based on complete data when both shape and scale parameters are unknown. The development is done by merging weight into GE to produce a new loss function called the weighted General Entropy loss function (WGE). Then, we utilized WGE to derive the parameters of the WD. After, we compared the performance of the developed estimation in this work with the Bayesian estimator using the GE loss function. Bayesian estimator using square error (SE) loss function, Ordinary Least Squares Method (OLS), Weighted Least Squared Method (WLS), and maximum likelihood estimation (MLE). Based on the Monte Carlo simulation method, those estimators are compared depending on the mean squared errors (MSE’s). The results show that the performance of the Bayes estimator under developed method (WGE) loss function is the best for estimating shape parameters in all cases and has good performance for estimating scale parameter.


Introduction
*The Weibull distribution (WD) is one of the most popular and widely used. Distributions in life testing and reliability studies. The Weibull distribution is mainly used in forecasting failure rates in numerous applications in various areas; for example, survival analysis, animal bioassay, breaking strength, and life expectancy.
Many researchers estimated the parameters of WD using several methods, including Bayesian and non-Bayesian (Aslam et al., 2014;Al-Duais and Alhagyan, 2020;Pobočíková and Sedliačková, 2014;Gupta and Singh, 2017;Guure et al., 2014;Mohammed and Ibrahim, 2011;Basumatary et al., 2005;Nwobi and Ugomma, 2014;Marks, 2005;Saeed et al., 2019;. In this paper, we will derive a Bayesian estimator under weighted General Entropy loss function (WGE) to estimate parameters of the Weibull distribution based on complete data. After, we will compare the proposed estimator with others.

Classical methods of estimation of Weibull parameters
The Classical methods selected for the comparative study are (i) Maximum Likelihood Estimator (MLE), (ii) Ordinary Least Squares Method (OLS), (iii) Weighted Least Squared Method (WLS).

Let
= 1 , 2 , 3 , … be the lifetime of a random sample of size drawn independently from the WD defined by 1, then the likelihood function for the given sample observations: Taking the natural logarithm of both sides yields, Differentiating (5) with respect to and in turn and equating to zero, we obtain the equations as follows: Unfortunately, there is no closed solution for Eqs. 6 and 7. Thus, as an alternative, numerical techniques are used like the Newton-Raphson method and the iteration method.
Eqs. 6 and 7 are nonlinear, so we will use the numerical analysis by the Newton-Raphson method to find the estimate of the parameters and .
Eq. 9 represents a simple linear regression function corresponding to ( ; , ): and ̂ it is a point estimator of ( ( ) ; , ), many estimators for ̂ are used, for example, the mean rank estimator ̂= Therefore, the estimates ̂ and ̂ of the parameters, and are given by, The estimates ̂ and ̂ of the parameters, and are given by,

Weighted least squared method (WLS)
The weighted least-squares (WLS) estimate of the parameters, , and are the values of the parameters which minimizes the function, The major difficulty in applying the WLS method is in finding in Eq. 14. To obtain the weights (Hung and Liu, 2004) used the delta method and obtained, Hence, the weights can be taken to be as follows: Minimizing * ( , ) we obtain the WLS estimates of , and are: where,

Loss function
The next subsections present three main types of loss function under study in this work.

Squared error loss function (SE)
Under the squared error loss function with the following form: The Bayes estimator of , denoted by ̂ is given by, ̂= ( | ).

General entropy loss function (GELF)
The General Entropy loss function for can be expressed as the following form (Calabria and Pulcini, 1996): where, ̂ is an estimate of . The Bayes estimator of , denoted by ̂ is the value ̂ which minimizes Eq. 21 and given as: provided that ( − ) exists and finite.

Weighted general entropy loss function (WGELF)
The researcher proposes this loss function depending on weighted loss function General Entropy as following: where represents the estimated parameter that makes the expectation of loss function (Eq. 23) as smallest as possible. While, ( ) represents the proposed weighted function, which equals to: Depending on the posterior distribution of the parameter , and by using the proposed weighted function as in Eq. 24, we can get the estimated weighted Bayes of the parameter as the following: So, we can find that: Consequently, the Bayesian estimation of the parameter using General. Entropy loss function will be: Provided that ( − ) and ( −( + ) ) exists and finite, where denotes the expected value. Note that, WGE loss function is a generalizing of the GE loss function, where GE is a special case of WGE when = 0 in Eq. 25.

Bayesian methods of estimation of Weibull parameters
In this section, we derive the Bayes estimates of the scale and shape parameters of WD, and we assume the Jeffrey formula as a prior distribution for each parameter as follow (Sinha, 1986): Therefore, the prior joint distribution function for parameters that we need to estimate are given as follow: By combining the likelihood function in Eq. 4 with the prior (p.d.f) of and in Eq. 28, then the posterior distribution of and is given by, where, The marginal posterior p.d.f for parameter can be obtained by integrating Eq. 19 with respect to as follow: The marginal posterior p. d. f for parameter can be obtained by integrating Eq. 29 with respect to as follow:

Estimates based on squared error loss function (SELF)
By using the Squared Error Loss function in Eq. 20, the Bayes estimator ̂ for , is given by,

Estimates based on general entropy loss function (GELF)
By using the General Entropy Loss function in Eq. 21, the Bayes estimator ̂ for , is given by, and the Bayes estimator ̂ for , is given by,

Estimates based on weighted general entropy loss function (WGELF)
Under the weighted General Entropy loss function, and by using 23, the Bayes estimator ̂ for is given by, where, and, Thus the weighted Bayes estimator for the shape parameter is:

Simulation
In this section, Monte-Carlo simulation is employed to compare the performance of four estimates (MLE, OLS, WOLS, and Bayes Estimators under different loss function including SE, GE, and WGE) for unknown shape and scale parameters based on the mean squared errors (MSE's) as follows: Step ( Step (2): Generate the sample random values of WD by using the formula: where u=RND(1) is a random variable distributed as a uniform distribution for the period (0,1).
Step (3): Calculating the ML, OLS. WOLS and Bayesian estimator of the parameters function of (WD) according to the formulas that had been obtained we calculate = (̂ ,̂).
Step (4): Comparing the different estimation methods according to the values of mean squared error (MSE), where: where ̂ is the estimate at the ℎ run.
The simulation results of MSE are tabulated in Tables 1-2.

Concluding remarks
From the results in the above Tables 1 and 2, we can state the following points: 1. Table 1 shows that the performance of the Bayesian estimator of the shape parameter under WGE loss is the best estimator comparing to the other estimators for all cases of ( , ) and all sample sizes . On the other hand, the performance of the WLS estimator of the shape parameter is the best estimator comparing to the MLE or OLS. The results also show that MSE's of all estimators of shape parameter is decreasing with the increase of the value of the scale parameter. 2. Table 2 shows that the performance of the Bayesian estimator of the scale parameter under WGE loss function with ( = 0.5 , = 0.5) ( = 1.5 , = 0.5) and ( = 1.5 , = 2.5) is the best estimator comparing to the other estimators for most sample sizes. While the performance of the Bayesian estimator of the scale parameter under GE loss function with ( = 0.5 , = 1.5) and ( = 1.5 , = 1.5) is the best estimator comparing to the other estimators for all sample sizes. On the other hand, the performance of the MLE estimator of the scale parameter with ( = 0.5 , = 0.5) and ( = 1.5 , = 0.5) is the best estimator comparing to the OLS or WLS, While the performance of the WLS estimator, with ( = 0.5 , = 1.5) ( = 1.5 , = 1.5) and ( = 1.5 , = 2.5) is the best estimator comparing to the MLE or OLS. The results also show that MSE's of all estimates of scale parameter is increasing for an increase of the Parameter value with all sample sizes. 3. The results showed that the values of all MSE's decrease as increases.

Conclusion
In this work, we developed the GE loss function to estimate the parameters of WD. The development was through merging a weight into GE to produce a new loss function called WGE. Then WGE was used to derive parameters of the WD.
Furthermore, we conducted a Monte Carlo simulation to examine the performance of the proposed method WGE. Then we compared the proposed method with other methods, including SE, GE, MLE, OLS, and WLS. The results show that the performance of the Bayes estimator under developed method (WGE) loss function is the best for estimating shape parameters in all cases and has good performance for estimating scale parameter.