Bootstrap Lower Confidence Limits of Superstructure Process Capability Indices for Esscher-Transformed Laplace Distribution

Abstract

Abstract This article is a comparative study between the parametric asymptotic lower confidence limits and bootstrap lower confidence limits for the basic quantile based process capability indices based on the unified super-structure C N p ⁢ ( u , v ) {C_{N_{p}}(u,v)} when the distribution of the quality characteristic follows an asymmetric non-normal distribution. We illustrate this method when the distribution of the quality characteristic is a member of the family of Esscher-transformed Laplace models introduced by S. George and D. George [11]. We obtain the bias corrected and accelerated (BCa) bootstrap confidence intervals of C N p ⁢ ( u , v ) {C_{N_{p}}(u,v)} , which provide lower confidence intervals with coverage probability nearer to the nominal value compared to the asymptotic confidence intervals. We conclude that for asymmetric and peaked processes, the BCa confidence interval is a better alternative compared to the usual confidence intervals under the assumption that the quality characteristic follows a Gaussian type distribution. Numerical examples are given based on some real data.