Estimation and confidence intervals of a new loss based process capability index ${\mathcal {C}}^{\prime }_{pm}$ with applications

Abstract

The process capability indices (PCIs) are frequently adopted to measure the performance of a process within the specifications. Although higher PCIs indicate higher process “quality”, yet it does not ascertain fewer rates of rejection. Thus, it is more appropriate to adopt a loss-based PCI for measuring the process capability. In this paper, our first objective is to introduce a new capability index called ${\mathcal {C}}^{\prime }_{pm}$ which is based on asymmetric loss function (linear exponential) for normal process which provides a tailored way of incorporating the loss in capability analysis. Next, we estimate the PCI $\mathcal C^{\prime }_{pm}$ when the process follows the normal distribution using six classical methods of estimation and compare the performance of the considered methods of estimation in terms of their mean squared errors through simulation study. Besides, four bootstrap methods are employed for constructing the confidence intervals for the index ${\mathcal {C}}^{\prime }_{pm}$. The performance of the bootstrap confidence intervals (BCIs) are compared in terms of average width and coverage probabilities using Monte Carlo simulation. Finally, for illustrating the effectiveness of the proposed methods of estimation and BCIs, two real data sets from electronic industries are analyzed. Simulation results show that ${\mathcal {P}}$-boot provides higher coverage probability for almost all sample sizes and for all the considered methods of estimation. In addition, among all considered methods of estimation, maximum product spacing estimator is the best as it produces the least width of the estimates. Further, real data analysis shows that width of bias-corrected accelerated bootstrap confidence interval is minimum among all other considered BCIs.