Point estimation of the process capability index Sₚₘₖ^ for the generalized KM exponential model with applications

In this paper, we conduct statistical inferences on the process capability index Sₚₘₖ for a lifetime distribution called the generalized KM exponential distribution. Some statistical properties of the distribution are reported, accompanied by figures that illustrate its shape characteristics based on the probability density function and the hazard rate function. The process capability index Sₚₘₖ is used to evaluate processes that deviate from a non-normal distribution. The maximum likelihood estimate for Sₚₘₖ^ is obtained using the invariance property of the maximum likelihood estimator. Furthermore, point estimates for Sₚₘₖ^^ are derived based on least squares, weighted least squares, Anderson-Darling, and Cramér-von Mises estimators. Furthermore, two real data analyses are performed to assess the applicability of the Sₚₘₖ^^ process capability index to the generalized KM exponential distribution.