Point estimation of the process capability index $ S_{pmk}^{\prime} $ for the generalized KM exponential model with applications

Year 2025, Volume: 74 Issue: 3, 478 - 491, 23.09.2025

Kadir Karakaya , Şule Sağlam

https://doi.org/10.31801/cfsuasmas.1592434

Abstract

In this paper, we conduct statistical inferences on the process capability index $S_{pmk}^{\prime }$ 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 $S_{pmk}^{\prime }$ is used to evaluate processes that deviate from a non-normal distribution. The maximum likelihood estimate for $S_{pmk}^{\prime}$ is obtained using the invariance property of the maximum likelihood estimator. Furthermore, point estimates for $S_{pmk}^{^{\prime}}$ 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 $S_{pmk}^{^{\prime}}$ process capability index to the generalized KM exponential distribution.

Keywords

Process capability index , lifetime distribution , maximum likelihood estimation , quality control

References

• Chen, J. P., Ding, C. G., A new process capability index for non-normal distributions, Int. J. Qual. Reliab. Manag., 18(7) (2001), 762-770.
• Choi, B. C., Owen, D. B., A study of a new process capability index, Commun. Stat. Theory Methods, 19(4) (1990), 1231-1245.
• Cordeiro, G. M., Ortega, E. M., Lemonte, A. J., The exponential–Weibull lifetime distribution, J. Statist. Comput. Simul., 84(12) (2014), 2592-2606.
• Erfanian, M., Sadeghpour Gildeh, B., A new capability index for non-normal distributions based on linex loss function, Quality Engineering, 33(1) (2021), 76-84.
• Juran, J. M., Godfrey, A. B., The Quality Control Process, McGraw-Hill, New York, 1999.
• Kane, V. E., Process capability indices, J. Qual. Technology, 18(1) (1986), 41-52.
• Kavya, P., Manoharan, M., A new lifetime model for non-monotone failure rate data, J. Indian Soc. Probab. Stat., 24(2) (2023), 211-241.
• Kumar, S., Yadav, A. S., Dey, S., Saha, M., Parametric inference of generalized process capability index Cpyk for the power Lindley distribution, Qual. Technol. Quant. M., 19(2) (2022), 153-186.
• Leiva, V., Marchant, C., Saulo, H., Aslam, M., Rojas, F., Capability indices for Birnbaum–Saunders processes applied to electronic and food industries, J. Appl. Stat., 41(9) (2014), 1881-1902.
• Lu, W., Shi, D., A new compounding life distribution: the Weibull–Poisson distribution, J. Appl. Stat., 39(1) (2012), 21-38.
• Maiti, S. S., Saha, M., Nanda, A. K., On generalizing process capability indices, Qual. Technol. Quant. M., 7(3) (2010), 279-300.
• Pappas, V., Adamidis, K., Loukas, S., A family of lifetime distributions, J. Qual. Reliab. Eng., 2012 (2012), Article ID 760687.
• Saha, M., Dey, S., Yadav, A. S., Kumar, S., Classical and Bayesian inference of Cpy for generalized Lindley distributed quality characteristic, Qual. Reliab. Eng. Int., 35(8) (2019), 2593-2611.
• Taguchi, G., A tutorial on quality control and assurance–the Taguchi methods, ASA Annual Meeting, Las Vegas, 1985.