On Scale Parameter Monitoring of the Rayleigh Distributed Data Using a New Design

Abstract

In recent years, many supplementary designs have been developed incorporating the assumption that data follow the particular non-normal distribution. The $V_{R}$ -control chart is one such design proposed to monitor the parameter $\sigma^{2}$ of the single parameter Rayleigh distributed data. Commonly, in authentic situations, practitioners need to estimate the scale parameter $\sigma $ in the observed processes instead of $\sigma ^{2}$ . However, the positive square root of the $V_{R}$ -statistic used in the existing design of $V_{R}$ -control chart is not an unbiased estimator of $\sigma $ and thus could not be practiced to monitor the scale parameter $\sigma $ of the Rayleigh distributed process. A new structure of the $V_{R}$ -control chart namely $V_{SQR}$ for monitoring the scale parameter of the Rayleigh distributed data has been originally developed in this study. The statistical basis of this newly $V_{SQR}$ design in terms of average run length $\left ({ARL }\right)$ , characteristic function and power curve have been derived. The analytical results are utilized further to determine the parameters of $V_{SQR}$ -chart and in comparing the performance of the proposed control chart with existing competitors. Comparative results illustrate the effectiveness of the proposed design in view of statistical power. Finally, the computational procedure of this newly $V_{SQR}$ -chart has been demonstrated using simulated data and real data on the breaking strength of carbon fibers.

Keywords

• Rayleigh distribution
• average run length
• power function
• unbiased estimator
• variable transformation