Process capability analysis for bounded measurements via the $S_{pmk}$ index

This study introduces a new flexible bounded distribution, namely the ratio-transformed Kumaraswamy distribution, to model data restricted to the unit interval $(0,1)$. Several main properties of the proposed distribution are derived, including the quantile function, moments, Lorenz and Bonferroni curves, order statistics, etc. The unknown parameters of the ratio-transformed Kumaraswamy distribution are estimated using maximum likelihood, least squares, weighted least squares, Anderson–Darling and Cramér–von Mises methods, and their finite-sample performances are evaluated through an extensive Monte Carlo simulation study based on bias, mean squared error, average absolute bias, and mean relative error criteria. The practical applicability of the proposed model is illustrated using two real datasets and compared with well-known bounded distributions such as the beta and Kumaraswamy distributions via several goodness-of-fit measures. Furthermore, the study extends the application of the ratio-transformed Kumaraswamy distribution to statistical quality control by adapting the process capability index $S_{pmk}$ to bounded measurements, deriving point and interval estimators, and assessing their performance through Monte Carlo simulation. The results demonstrate that the ratio-transformed Kumaraswamy distribution offers increased flexibility and improved modeling capability for bounded data, providing an effective alternative for process capability analysis in quality control applications.