This paper deals with making inferences regarding system reliability R = P(X < Y ) when the distribution of the stress X and the strength Y are independent Weibull. In the literature, estimators based on simple random sampling (SRS) are widely used in estimating R. However, in recent years, ranked set sampling (RSS) has become popular in performing statistical inference. We, therefore, obtain the estimators of R based on RSS using maximum likelihood (ML) and modified maximum likelihood (MML) methodologies. The performances of the proposed estimators are compared with their counterparts based on SRS using Monte Carlo simulation. The simulation results show that the proposed estimators are more preferable than the estimators based on SRS in terms of efficiency. In addition, under the assumption of imperfect ranking the efficiencies of the ML and the MML estimators of R, based on RSS, are compared and the ML estimator of R is found to be more efficient. Finally, a real data-set is analysed to demonstrate the implementation of the proposed estimators at the end of the paper.
Inference for the system reliability R is one of the most popular problems in the areas of engineering, statistics, biostatistics and etc. Therefore, there exist considerable numbers of studies concerning this problem. Traditionally, simple random sampling (SRS) is used for estimating the system reliability. However, in recent years, ranked set sampling (RSS), cost effective and efficient alternative of SRS, is used to estimate the system reliability. In this study, we consider the interval estimation of R when both the stress and the strength are independent Weibull random variables based on RSS. We first obtain the asymptotic confidence interval (ACI) of R by using the maximum likelihood (ML) methodology. The bootstrap confidence interval (BCI) of R is also constructed as an alternative to ACI. An extensive Monte-Carlo simulation study is conducted to compare the performances of ACI and BCI of R for different settings. Finally, a real data set is analyzed to demonstrate the implementation of the proposed methods.
In statistical literature, estimation of = (<) is a commonly-investigated problem, and consequently, there have been considerable number of studies dealing with its estimation of it under simple random sampling (SRS). However, in recent years, the ranked set sampling (RSS) method have been widely-used in the estimation of. In this study, we consider the estimation of when the distribution of the both stress and strength are Weibull under the modification of RSS, which are extreme ranked set sampling (ERSS), median ranked set sampling (MRSS) and percentile ranked set sampling (PRSS). We obtain the estimators of using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies under these modifications. Then the performances of proposed estimators are compared with the corresponding ML and MML estimators of using SRS via a Monte-Carlo simulation study.
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