Parametric inference of generalized process capability index Cpyk for the power Lindley distribution

Abstract

ABSTRACT In this article, to estimate the generalized process capability index (GPCI) Cpyk when the process follows the power Lindley distribution, we have used five methods of estimation, namely, maximum likelihood method of estimation, ordinary and weighted least squares method of estimation, the maximum product of spacings method of estimation, and Bayesian method of estimation. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential) loss functions with the help of the Metropolis-Hastings algorithm and importance sampling method. The confidence intervals for the GPCI Cpyk is constructed based on three bootstrap methods and Bayesian methods. Besides, asymptotic confidence intervals based on maximum likelihood method is also constructed. We studied the performances of these estimators based on their corresponding biases and MSEs for the point estimates of GPCI Cpyk, and coverage probabilities (CPs), and average width (AW) for interval estimates. It is found that the Bayes estimates performed better than the considered classical estimates in terms of their corresponding MSEs. Further, the Bayes estimates based on linear-exponential loss function are more efficient than the squared error loss function under informative prior. To illustrate the performance of the proposed methods, two real data sets are analyzed.In this article, to estimate the generalized process capability index (GPCI) when the process follows the power Lindley distribution, we have used five methods of estimation, namely, maximum likelihood method of estimation, ordinary and weighted least squares method of estimation, the maximum product of spacings method of estimation, and Bayesian method of estimation. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential) loss functions with the help of the Metropolis-Hastings algorithm and importance sampling method. The confidence intervals for the GPCI is constructed based on three bootstrap methods and Bayesian methods. Besides, asymptotic confidence intervals based on maximum likelihood method is also constructed. We studied the performances of these estimators based on their corresponding biases and MSEs for the point estimates of GPCI , and coverage probabilities (CPs), and average width (AW) Abbreviations: AW : Average width; : Bias-corrected percentile bootstrap; BCI : Bootstrap confidence interval; CDF : Cumulative distribution function; CI : Confidence interval; CK : Coefficient of kurtosis; CP : Coverage probability; CS : Coefficient of skewness; GGD : Generalized gamma distribution; GPCI : Generalized process capability index; GLD : Generalized lindley distribution; SWCI : Shortest width credible interval; IS : Importance sampling; K-S : Kolmogorov-Smirnov; : Lower specification limi; LD : Lindley distribution; LDL : Lower desired limitLLF : Linex loss function; MCMC : Markov Chain Monte Carlo; MH : Metropolis-Hastings; MPSE : Maximum product of spacings estimator; MLE : Maximum likelihood estimator; MSE : Mean squared error; OLSE : Ordinary least squares estimator; : Percentile bootstrap; PDF : Probability density function; PCI : Process capability index; PLD : Power Lindley distribution; : -th quartile; SD : Standard deviation; : Standard bootstrap; SELF : Squared error loss function; : Target value; : Upper specification limit; : Upper desired limit; WD : Weibull distribution; WLSE : Weighted least squares estimator