A New Asymmetric Loss-Based Process Capability Index and Its Applications to Industrial Data

This paper introduces a new capability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(CQT)} which is based on an asymmetric loss function named Quadratic-Type Squared Error Loss Function (QTSELF). To estimate the proposed index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> </m:math> {CQT} , we have used five different classical methods of estimation (namely maximum likelihood, least squares and weighted least squares, Cramér–von-Mises, Anderson–Darling and maximum product of spacing) and their performances are compared. Additionally, three bootstrap confidence intervals, namely standard bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> {S} -boot), percentile bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒫</m:mi> </m:math> {P} -boot) and Student’s t bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {T} -boot) are constructed. Their performances are compared in terms of average widths and coverage probabilities through a Monte-Carlo simulation study. Finally, two datasets are taken from electronic industries to demonstrate the practical utilities of the proposed estimation techniques and confidence intervals.