Bootstrap Confidence Intervals of Process Capability Indices <i>S</i> _<i>pmk</i> and <i>C</i> _<i>py</i> Using Different Methods of Estimation for Marshall‐Olkin Inverse Log‐Logistic Distribution

ABSTRACT In this study, we propose six classical estimation methods, including maximum likelihood, least squares, weighted least squares, Cramr‐von‐Mises, Anderson–Darling, and right‐tail Anderson–Darling for estimating the unknown parameters of the process capability indices and for the Marshall–Olkin inverse log‐logistic distribution. Additionally, we developed the Bayesian estimators of the unknown parameters of and of the Marshall–Olkin inverse log‐logistic distribution using Lindley's approximation for both symmetric and asymmetric loss functions, namely squared error loss function and LINEX (linear exponential) loss function, respectively. Furthermore, we considered bootstrap confidence intervals for obtaining confidence intervals of the process capability indices employing the aforementioned estimation methods. The performance of the process capability indices is assessed using simulation study for varying sample sizes in terms of coefficients of skewness, kurtosis, mean squared error, and bootstrap confidence interval widths. It is concluded that the Bayesian estimator under an asymmetric loss function performs better than its competitors. This study demonstrates the practical relevance of research findings by addressing three real‐life data applications, highlighting their tangible impact.