Estimation of a New Asymmetry Based Process Capability Index 𝐶𝑐 for Gamma Distribution

Sanku Dey; Mahendra Saha

doi:10.1515/eqc-2024-0029

Enjoy 40% off

academic books on De Gruyter Brill *

Article

Estimation of a New Asymmetry Based Process Capability Index 𝐶_𝑐 for Gamma Distribution

Sanku Dey and Mahendra Saha

Published/Copyright: December 5, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Stochastics and Quality Control Volume 40 Issue 1

Abstract

In this work, we provide five different classical estimation methods for the gamma distribution and unknown parameters, which are used to estimate the newly constructed process capability index (PCI) C c . Next, we use the aforementioned estimation techniques to consider five bootstrap confidence intervals: standard bootstrap (SB), percentile bootstrap (PB), student’s 𝑡 bootstrap (STB), bias-corrected percentile bootstrap (BCPB), and bias-corrected accelerated bootstrap (BCAB). These allow us to construct confidence intervals of the PCI C c . Using the suggested techniques of estimate for small, moderate, and large sample sizes, Monte Carlo simulations are run to compare the performances of the bootstrap confidence intervals’ average widths and coverage probabilities. Simulation results indicate that MLE gives the least AW for all BCIs (SB, PB, BCPB, TB, and BCAB) when compared to all other estimation approaches evaluated in this paper. Further, simulation results demonstrate that, for almost all sample sizes and all of the examined estimation approaches, BCAB CI generates smaller AW and larger CP than their counterparts. Finally, for illustrative purposes, three real data sets are analysed.

Keywords: Bootstrap Confidence Intervals; Classical Methods of Estimation; Measure of Skewness; Process Capability Index; Gamma Distribution

MSC 2020: 62E15; 62F10; 62F12; 62F40; 62F25

Funding source: University Grants Commission

Award Identifier / Grant number: /IoE/2024-25/12/FRP

Funding statement: This work was supported by the IoE [Ref. No. /IoE/2024-25/12/FRP], Dept. of Statistics, Faculty of Mathematical Sciences, University of Delhi.

Acknowledgements

The authors thank the editor and the reviewer for their very careful reading and constructive comments which helped us to improve the earlier version of this article.

References

[1] L. K. Chan, S. W. Cheng and F. A. Spiring, A new measure of process capability: C p ⁢ m , J. Qual. Technol. 30 (1988), 162–175. 10.1080/00224065.1988.11979102Search in Google Scholar

[2] J. P. Chen and C. G. Ding, A new process capability index for non-normal distributions, Int. J. Qual. Reliab. Manag. 18 (2001), no. 7, 762–770. 10.1108/02656710110396076Search in Google Scholar

[3] R. C. H. Cheng and N. A. K. Amin, Estimating parameters in continuous univariate distributions with a shifted origin, J. R. Stat. Soc. Ser. B. Stat. Methodol. 3 (1983), 394–403. 10.1111/j.2517-6161.1983.tb01268.xSearch in Google Scholar

[4] Y.-M. Chou, A. M. Polansky and R. L. Mason, Transforming non-normal data to nor-mality in statistical process control, J. Qual. Technol. 30 (1998), 133–141. 10.1080/00224065.1998.11979832Search in Google Scholar

[5] J. A. Clements, Process capability calculations for non-normal distributions, Qual. Progress 22 (1989), 95–100. Search in Google Scholar

[6] M. H. de Almeida, P. L. Ramos, G. S. Rao and F. A. Moala, Objective Bayesian inference for the capability index of the Gamma distribution, Qual. Reliab. Eng. Int. 37 (2021), no. 5, 2235–2247. 10.1002/qre.2854Search in Google Scholar

[7] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Non-Linear Equations, Prentice-Hall, Englewood Cliffs, 1983. Search in Google Scholar

[8] S. Dey and M. Saha, Bootstrap confidence intervals of generalized process capability index C p ⁢ y ⁢ k using different methods of estimation, J. Appl. Stat. 46 (2019), no. 10, 1843–1869. 10.1080/02664763.2019.1572721Search in Google Scholar

[9] S. Dey and M. Saha, Bootstrap confidence intervals of process capability index S p ⁢ m ⁢ k using different methods of estimation, J. Stat. Comput. Simul. 90 (2020), 28–50. 10.1080/00949655.2019.1671980Search in Google Scholar

[10] S. Dey, M. Saha, M. Z. Anis, S. S. Maiti and S. Kumar, Estimation and confidence intervals of C N ⁢ p ⁢ ( u , v ) for logistic-exponential distribution with application, Int. J. System Assurance Eng. Manag. (2023), 10.1007/s13198-023-01870-y. 10.1007/s13198-023-01870-ySearch in Google Scholar

[11] S. Dey, M. Saha and C. Park, Applications of a new loss and cost-based process capability index to electronic industries, Commun. Statist. Case Stud. Data Anal. Appl. 9 (2023), no. 4, 351–377. 10.1080/23737484.2023.2249850Search in Google Scholar

[12] S. Dey, L. Wang and M. Saha, Confidence intervals of S p ⁢ m ⁢ k ′ based on bias-corrected methods of estimation for generalized exponential distribution, Int. J. System Assurance Eng. Manag. (2024), 10.1007/ s13198-024-02533-2. 10.1007/ s13198-024-02533-2Search in Google Scholar

[13] S. Dey, C. Zhang and M. Saha, Classical and Bayesian estimation of the index C p ⁢ m ⁢ k and its confidence intervals for normally distributed quality characteristic, J. Stat. Comput. Simul. 91 (2021), no. 10, 1911–1934. 10.1080/00949655.2021.1879079Search in Google Scholar

[14] J. J. Flaig, Process capability sensitivity analysis, Qual. Eng. 11 (1999), 587–592. 10.1080/08982119908919279Search in Google Scholar

[15] T. C. Hsiang and G. Taguchi, A tutorial on quality control and assurance, Annual Meeting on the American Statistical Association, unpublished presentation, Las Vegas (1985). Search in Google Scholar

[16] R. Ihaka and R. Gentleman, R: A language for data analysis and graphics, J. Comput. Graph. Statist. 5 (1996), 299–314. 10.1080/10618600.1996.10474713Search in Google Scholar

[17] P. C. James, C p ⁢ k equivalencies, Quality 28 (1989), no. 9, Paper No. 75. Search in Google Scholar

[18] J. M. Juran, Juran’s Quality Control Handbook, 3rd ed., McGraw-Hill, New York, 1974. Search in Google Scholar

[19] V. E. Kane, Process capability indices, J. Qual. Technol. 18 (1986), 41–52. 10.1080/00224065.1986.11978984Search in Google Scholar

[20] T. Kanwal and K. Abbas, Bootstrap confidence intervals of process capability indices Spmk, Spmkc and Cs for Frechet distribution, Qual. Reliab. Eng. Int. 39 (2023), 2244–2257. 10.1002/qre.3333Search in Google Scholar

[21] M. Kashif, M. Aslam, A. H. Al-Marshadi and C. H. Jun, Capability indices for non-normal distribution using Gini’s mean difference as measure of variability, IEEE Access 4 (2016), 7322–7330. 10.1109/ACCESS.2016.2620241Search in Google Scholar

[22] M. Kashif, M. Aslam, G. S. Rao, A. H. Al-Marshadi and C. H. Jun, Bootstrap confidence intervals of the modified process capability index for Weibull distribution, Arab. J. Sci. Eng. 42 (2017), 4565–457. 10.1007/s13369-017-2562-7Search in Google Scholar

[23] V. Leiva, C. Marchant, H. Saulo, M. Aslam and F. Rojas, Capability indices for Birnbaum–Saunders processes applied to electronic and food industries, J. Appl. Stat. 41 (2014), 1881–1902. 10.1080/02664763.2014.897690Search in Google Scholar

[24] A. Luciano, A process capability index with reliable confidence intervals, Comm. Statist. Simulation Comput. 25 (1996), 235–245. 10.1080/03610919608813310Search in Google Scholar

[25] S. S. Maiti, M. Saha and A. K. Nanda, On generalizing process capability indices, J. Qual. Technol. Quant. Manag. 7 (2010), no. 3, 279–300. 10.1080/16843703.2010.11673233Search in Google Scholar

[26] W. L. Pearn, S. Kotz and N. L. Johnson, Distributional and inferential properties of process capability indices, J. Qual. Technol. 24 (1992), 216–231. 10.1080/00224065.1992.11979403Search in Google Scholar

[27] W. L. Pearn, Y. T. Tai, I. F. Hsiao and Y. P. Ao, Approximately unbiased estimator for non-normal process capability index C N ⁢ p ⁢ k , J. Testing Evaluation 42 (2014), 1408–1417. 10.1520/JTE20130125Search in Google Scholar

[28] W. L. Pearn, Y. T. Tai and H. T. Wang, Estimation of a modified capability index for non-normal distributions, J. Testing Evaluation 44 (2016), 1998–2009. 10.1520/JTE20150357Search in Google Scholar

[29] C. Peng, Parametric lower confidence limits of quantile-based process capability indices, J. Qual. Technol. Quant. Manag. 7 (2010), no. 3, 199–214. 10.1080/16843703.2010.11673228Search in Google Scholar

[30] C. Peng, Estimating and testing quantile-based process capability indices for processes with skewed distributions, J. Data Sci. 8 (2010), no. 2, 253–268. 10.6339/JDS.2010.08(2).582Search in Google Scholar

[31] G. P. Phillips, Target ratio simplifies capability index system, makes it easy to use C p ⁢ m , Qual. Eng. 7 (1994), 299–313. 10.1080/08982119408918785Search in Google Scholar

[32] M. R. Pina-Monarrez, J. F. Ortiz-Yañez and M. I. Rodríguez-Borbón, Non-normal capability indices for the Weibull and log-normal distributions, Qual. Reliab. Eng. Int. 32 (2016), 1321–1329. 10.1002/qre.1832Search in Google Scholar

[33] M. Saha, M. Bhattacharya, S. Pramanik, S. S. Maiti and A. Gupta, A unified generalized process capability index and its estimation for logistic-exponential distributed characteristic, Eur. J. Indust. Eng., to appear. Search in Google Scholar

[34] M. Saha and S. Dey, Confidence intervals of a loss based PCI S p ⁢ m ⁢ k ′ using exponentiated-exponential distributed quality characteristics, Amer. J. Math. Manag. Sci. 1 (2023), 10.1080/01966324.2023.2175632. 10.1080/01966324.2023.2175632Search in Google Scholar

[35] A. Sarkar and S. Pal, Estimation of process capability index for concentricity, Qual. Eng. 9 (1997), 665–671. 10.1080/08982119708919088Search in Google Scholar

[36] M. Smithson, Correct confidence intervals for various regression effect sizes and parameters: The importance of non-central distributions in computing intervals, Educational Psychol. Measurement 61 (2001), 605–632. 10.1177/00131640121971392Search in Google Scholar

[37] S. E. Sommerville and D. C. Montgomery, Process capability indices and non-normal distributions, Qual. Eng. 9 (1996), 305–316. 10.1080/08982119608919047Search in Google Scholar

[38] J. Swain, S. Venkatraman and J. Wilson, Least squares estimation of distribution function in Johnsons translation system, J. Stat. Comput. Simul. 29 (1988), 271–297. 10.1080/00949658808811068Search in Google Scholar

[39] L. C. Tang and S. E. Than, Computing process capability indices for non-normal data: A review and comparative study, Qual. Reliab. Eng. Int. 15 (1999), no. 5, 339–353. 10.1002/(SICI)1099-1638(199909/10)15:5<339::AID-QRE259>3.0.CO;2-ASearch in Google Scholar

[40] K. Vannman, A uniformed approach to capability indices, Statist. Sinica 5 (1995), no. 2, 805–820. Search in Google Scholar

[41] S. Weber, T. Ressurreio and C. Duarte, Yield prediction with a new generalized process capability index applicable to non-normal data, IEEE Trans. Comput.-Aided Design Integrated Circuits Syst. 35 (2016), 931–942. 10.1109/TCAD.2015.2481865Search in Google Scholar

[42] P. A. Wright, A process capability index sensitive to skewness, J. Stat. Comput. Simul. 52 (1995), 195–203. 10.1080/00949659508811673Search in Google Scholar

Received: 2024-08-04

Revised: 2024-10-28

Accepted: 2024-11-22

Published Online: 2024-12-05

Published in Print: 2025-05-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/eqc-2024-0029

Keywords for this article

Bootstrap Confidence Intervals; Classical Methods of Estimation; Measure of Skewness; Process Capability Index; Gamma Distribution