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

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) ${\mathcal{C}}_{pyk}$ 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 ${\mathcal{C}}_{pyk}$ 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 ${\mathcal{C}}_{pyk}$, and coverage probabilities (CPs), and average width (AW)

Abbreviations: AW : Average width; $\mathcal{P}\mathcal{B}$ : 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; $L$ : Lower specification limi; LD : Lindley distribution; LDL : Lower desired limit LLF : 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; $\mathcal{P}\mathcal{B}$ : Percentile bootstrap; PDF : Probability density function; PCI : Process capability index; PLD : Power Lindley distribution; $Q_i$ : $i$-th quartile; SD : Standard deviation; $\mathcal{S}\mathcal{B}$ : Standard bootstrap; SELF : Squared error loss function; $T$ : Target value; $U$ : Upper specification limit; $UDL$ : Upper desired limit; WD : Weibull distribution; WLSE : Weighted least squares estimator

KEYWORDS:

• Asymptotic confidence interval
• bootstrap confidence interval
• Metropolis-Hastings algorithm
• importance sampling method
• Monte Carlo simulation
• process capability index

Acknowledgments

The authors thank the editor, associate editor and referees for their comments and helpful suggestions which helped to improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Sumit Kumar

Sumit Kumar: He is currently working as an Assistant Professor in the Department of Engineering & Physical Aciences, IAR, Gandhinagar, Gujrat. He did his Ph.D. in Statistics from Central University of Rajasthan. He did his M.Sc. in Statistics (2014) from Chaudhary Charan Singh University, Meerut. He is a young researcher and he has published 8 research articles in reputed international journals.

Abhimanyu Singh Yadav

Abhimanyu Singh Yadav: He is currently working as an Assistant Professor in the Department of Statistics, Banaras Hindu University, India. He also worked as Assistant Professor in the Department of Statistics, PUC, Mizoram University from December 2014 to September 2017, Aizawl, India. He did his M.Sc. in Statistics (2011) from Banaras Hindu University, Varanasi and Ph.D. in Statistics (2015) from the same University. He has received merit scholarship in M. Sc. first year from the Department of Statistics, Banaras Hindu University. He is a young researcher and published 32 research papers in the area of Classical and Bayesian inference, reliability theory and distribution theory in national and international journals of repute. He reviewed more than 10 research papers for various well reputed national/international journals.

Sanku Dey

Sanku Dey: He is currently workings as an Associate Professor in the Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India. He did his M.Sc. in Statistics in the year of 1991 from Gauhati University, Guwahati, India and Ph.D. in Statistics (reliability theory) in the year 1998 from the same university. He has published more than 155 researchpapers in national and international journals of repute. He reviewed more than 300 research papers for various well reputed international journals. He is an associate editor of American Journal of Mathematical and Management Sciences and also the member of editorial board of several national and international journals of repute. He is a renowned researcher and has a good number of contributions in almost all fields of Statistics viz, distribution theory, discretization of continuous distribution, reliability theory, multicomponent stress-strength reliability, survival analysis, Bayesian inference, Record Statistics, Statistical quality control, order statistics, lifetime performance index based on classical and Bayesian approach as well as different types of censoring schemes etc.

Mahendra Saha

Mahendra Saha: He is currently workings as an Assistant Professor in the Department of Statistics, Central University of Rajasthan, India. He did his M.Sc. in Statistics (2007) from Vishva-Bharti University, West Bengal and Ph.D. in Statistics (2012) from the same University. He has received one major project (April 2015-March 2017) from UGC, New Delhi, Govt. of India. He has published two books in the area of statistical process control from Germany. Also, he has good contribution in the area of statistical quality control, reliability theory, and distribution theory and published around 38 research articles in reputed national and international journals. He reviewed more than 14 research papers for various well reputed national/international journals.