On Progressively Type-II Censored Two-Parameter Rayleigh Distribution

Abstract Recently, Rayleigh distribution has received considerable attention in the statistical literature. In this paper, we consider the point and interval estimation of the functions of the unknown parameters of a two-parameter Rayleigh distribution. First, we obtain the maximum likelihood estimators (MLEs) of the unknown parameters. The MLEs cannot be obtained in explicit forms, and we propose to use the maximization of the profile log-likelihood function to compute the MLEs. We further consider the Bayesian inference of the unknown parameters. The Bayes estimates and the associated credible intervals cannot be obtained in closed forms. We use the importance sampling technique to approximate (compute) the Bayes estimates and the associated credible intervals. For comparison purposes we have also used the exact method to compute the Bayes estimates and the corresponding credible intervals. Monte Carlo simulations are performed to compare the performances of the proposed method, and one data set has been analyzed for illustrative purposes. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. A data example has been provided for illustrative purposes

Implementing a Lifetime Performance Index of Products with a Two-Parameter Rayleigh Distribution Under a Progressively type II Right Censored Sample

In manufacturing, quality control is a process that ensures customers receive products free from defects and meet their needs. Process capability analysis has been widely applied in the field of quality control to find out how well a given process meets a set of specification limits. The lifetime performance index \begin {math} C_L ,\end {math} a type of process capability index is used to measure the larger-the-better type quality characteristics.  Under the assumption of Two-Parameter Rayleigh Distribution, this study constructs a maximum likelihood estimator of  \begin {math} C_L \end {math} based on the progressively type II right censored sample. The maximum likelihood estimator of  \begin {math} C_L \end {math} is then utilized to develop the new hypothesis testing procedure. The testing procedure can be employed the testing procedure to determine whether the lifetime of a product adheres to the required level