Estimation of a New Asymmetry Based Process Capability Index 𝐶_𝑐 for Gamma Distribution

Abstract In this work, we provide five different classical estimation methods for the gamma distribution and unknown parameters, which are used to estimate the newly constructed process capability index (PCI) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>c</m:mi> </m:msub> </m:math> C_{c} . Next, we use the aforementioned estimation techniques to consider five bootstrap confidence intervals: standard bootstrap (SB), percentile bootstrap (PB), student’s 𝑡 bootstrap (STB), bias-corrected percentile bootstrap (BCPB), and bias-corrected accelerated bootstrap (BCAB). These allow us to construct confidence intervals of the PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>c</m:mi> </m:msub> </m:math> C_{c} . Using the suggested techniques of estimate for small, moderate, and large sample sizes, Monte Carlo simulations are run to compare the performances of the bootstrap confidence intervals’ average widths and coverage probabilities. Simulation results indicate that MLE gives the least AW for all BCIs (SB, PB, BCPB, TB, and BCAB) when compared to all other estimation approaches evaluated in this paper. Further, simulation results demonstrate that, for almost all sample sizes and all of the examined estimation approaches, BCAB CI generates smaller AW and larger CP than their counterparts. Finally, for illustrative purposes, three real data sets are analysed.