Classical and Bayesian Estimation of PCI 𝒞 pc Using Power Generalized Weibull Distribution

This article focuses on estimating the process capacity index (PCI), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>pc</m:mi> </m:msub> </m:math> Cpc , where the underlying distribution is a power generalized Weibull distribution, using maximum likelihood and Bayesian techniques. The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>pc</m:mi> </m:msub> </m:math> Cpc index may be used to both normal and non-normal quality attributes. This article aims to accomplish three things: Using the maximum likelihood approach, we first determine the estimator of the PCI <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>pc</m:mi> </m:msub> </m:math> Cpc . Secondly, the Metropolis-Hastings Algorithm is used to study Bayesian estimation under four loss functions (symmetric and asymmetric). Third, the index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>pc</m:mi> </m:msub> </m:math> Cpc ’s 95 confidence intervals are built using Bayesian and four bootstrap techniques. The point estimates of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">𝒞</m:mi> <m:mi>pc</m:mi> </m:msub> </m:math> Cpc have been evaluated using Monte Carlo simulation in terms of mean square errors (MSEs), four bootstrap methods, and highest posterior density (HPD) credible intervals in relation to their average width (AW) and coverage probabilities (CPs). Using two real data sets – one pertaining to the size of electrical connections and the other to the amount of protein (in g) for adult patients at the Hospital Carlos Van Buren in Valparaiso, Chile – a comparable analysis to that employed in the simulations is conducted in order to illustrate the effectiveness of the suggested methods.