Also, it is to be mentioned that it is the first time to consider ten estimation methods of estimation to estimate the stress-strength reliability of the Weibull distribution. In addition, the simulation study and the real data analysis prove that there are classical methods rather than the MLE methods which can be give a desirable estimates, motivating its use in applied areas. Further, we develop a guideline for choosing the best estimation method to estimate the stress-strength reliability of the Weibull distribution, which we think would be of deep interest to applied statisticians/reliability engineers. In this paper, we focus on ten different estimation methods to estimate the stress-strength reliability and investigate how the different estimators behave for different sample sizes and for different parameter values. For this reason, our aim in this paper is to use nine frequentist methods of estimation besides the maximum likelihood estimation (MLE) method. But no much attention has been paid to use other frequentist methods to estimate R = P( Y < X), while in some cases they can provide better estimates than the maximum likelihood method. These studies considered the maximum likelihood as a frequentist method as well as the Bayesian estimation method. Many authors investigated the estimation of R = P( Y < X) when X and Y are independent Weibull random variables, see for example, and see the references cited therein. These studies have actually considered the maximum likelihood and Bayesian estimation methods under different life testing schemes for various distributions. In recent years, a many works have been done on the problem of estimating R of some distributions See for example, and. considered the estimation of R, where X and Y are two independent generalized Pareto distributions. obtained the estimation of R, where X and Y are distributed as two independent three parameter generalized exponential random variables. studied the estimation of R, where X and Y are generalized exponential and Weibull random variables, respectively. In the statistical literature, several authors considered the estimation problem of the stress strength parameter. In this case the system fails if and only if the applied stress is larger than its strength at any time. Let X and Y be two independent random variables showing the strength of a system and the stress applied to it, respectively, then R = P( Y < X) is the measure of the system performance which arises quite naturally in the mechanical reliability of a system. Since the pioneering work of Birnbaum, the statistical inference of the stress-strength parameter have received considerable attention and commonly used in many engineering applications. To show the applicability and the importance of the proposed estimators, we analyze one real data set. The Mont Carlo simulation study revels that the percentile and maximum product of spacing methods are highly competitive with the other methods for small and large sample sizes. The performance and the finite sample properties of the different estimators are compared in terms of relative biases and relative mean squared errors. We compare the efficiency of the different proposed estimators by conducting an extensive Mont Carlo simulation study. We also consider two parametric bootstrap confidence intervals of R. Other than the maximum likelihood method, a nine frequentist estimation methods are used to estimate R, namely: least square, weighted least square, percentile, maximum product of spacing, minimum spacing absolute distance, minimum spacing absolute-log distance, method of CramГ©r-von Mises, Anderson-Darling and Right-tail Anderson-Darling. The start point to estimate the parameter R is the maximum likelihood method. For the first time, ten frequentist estimation methods are considered on stress-strength reliability R = P( Y < X) when X and Y are two independent Weibull distributions with the same shape parameter.

Weibull distribution reliability