Percentile bootstrap control chart for monitoring process variability under non-normal processes

Document Type : Article

Authors

1 College of Statistical and Actuarial Sciences, University of the Punjab Lahore-54000, Lahore, Pakistan

2 Department of Statistics, GC University Faisalabad, Faisalabad, Pakistan

3 Department of Statistic, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia

Abstract

In the recent years, another approach named as the bootstrap method is getting popular in Statistical Process Control (SPC) specifically when the underlying distribution of the process is unknown. The bootstrap estimators are getting popularity in statistical process control due to their remarkable properties for non-normal distribution. In this paper the bootstrap control chart is developed for monitoring process variability and robustness is discussed through simulation studies. It appears that the proposed control chart for monitoring process variability based on the bootstrap method is performing better to detect out-of-control signal in a case when data follow skewed distributions. Therefore, the proposed chart is more recommendable for industrial practitioners.

Keywords


References:
1. Shewhart, W.A., Economic Control of Quality of Manufacturing Product, Van Nostrand, New York (1931). 
2. Liu, R.Y. and Tang, J. "Control charts for dependent and independent measurements based on bootstrap methods", Journal of the American Statistical Association, 91(436), pp. 1694-1700 (1996). DOI: 10.1080/01621459.1996.10476740.
3. Bonett, D.G. "Confidence interval for a coefficient of quartile variation", Computational Statistics & Data Analysis, 50, pp. 2953-2957 (2006). https://doi.org/10.1016/j.csda.2005.05.007.
4. Tukey, J.W. "A survey of sampling from contaminated distributions", In Contributions to Probability and Statistics, Essays in Honor of Harold Hotelling, Olkin I. et al. (Eds.), Stanford University Press: Stanford, pp. 448-485 (1960).
5. Mosteller, F. and Tukey, J.W., Data Analysis and Regression: A Second Course in Statistics, Addison- Wesley, USA (1977). 
6. Hampel, F.R. "The in uence curve and its role in robust estimation", Journal of the American Statistical Association, 69, pp. 383-393 (1974). https://doi.org/10.1080/01621459.1974.10482962.
7. Abu-Shawiesh, M.O. "A simple robust control chart based on MAD", Journal of Mathematics and Statistics, 4(2), pp. 102-107 (2008). DOI: 10.3844/jmssp.2008.102.107.
8. Adekeye, K.S. and Azubuike, P.I. "Derivation of the limits for control chart using the median absolute deviation for monitoring non normal process", Journal of Mathematics and Statistics, 8(1), pp. 37-41 (2012). DOI: 10.3844/jmssp.2012.37.41.
9. Rousseeuw, P.J. and Croux, C. "Alternative to median absolute deviation", Journal of the American Statistical Association, 88(424), pp. 1273-1283 (1993). DOI: 10.1080/01621459.1993.10476408.
10. Efron, B. "Bootstrap methods: another look at the Jackknife", The Annals of Applied Statistics, 7, pp. 1- 26 (1979). DOI: 10.1214/aos/1176344552.
11. Liu, Y., Liang, J., and Qian, J. "Moving blocks bootstrap control chart for dependent multivariate data", IEEE International Conference on Systems, Man and Cybernetics, pp. 5177-5182 (2004). DOI: 10.1109/ICSMC.2004.1401016.
12. Nichols, M.D. and Padgett, W.J. "A bootstrap control chart for weibull percentiles", Quality and Reliability Engineering International, 22, pp. 141-151 (2006). DOI: 10.1002/qre.691.
13. Chatterjee, S. and Qiu, P. "Distribution-free cumulative sum control charts using bootstrap based control limits", The Annals of Applied Statistics, 3(1), pp. 349-369 (2009). DOI: 10.1214/08-AOAS197.
14. Wararit, P. and Somchit, W. "Bootstrap confidence intervals of the di erence between two process capability indices for half logistic distribution", Pakistan Journal of Statistics and Operation Research, 8(4), pp. 879-894 (2012). DOI: 10.18187/pjsor.v8i4.455.
15. Saeed, N. and Kamal, S. "The bootstrap S-chart for process variability: an alternative to MAD chart", Journal of Quality and Technology Management, 10(2), pp. 109-124 (2014). https://pu.edu.pk/images/journal/iqtm/PDFFILES/ 05-The%20 Bootstrap%20SChart% 20for%20Process%20Variability- Nadia%20Saeed.pdf.
16. Wang, D. and Hryniewicz, O. "A fuzzy non-parametric shewhart chart based on the bootstrap approach", International Journal of Applied Mathematics and  Computer Science, 25(2), pp. 389-401 (2015). https://doi.org/10.1515/amcs-2015-0030.
17. Hila, Z.N., Safiih, L.M., and Shazrahanim, K.N. "Modeling of moving centerline exponentially weighted moving average (MCEWMA) with bootstrap approach", International Journal of Applied Business and Economics Research, 14(2), pp. 621-638 (2016). https://serialsjournals.com/abstract/62286 621- 638.pdf.
18. Zhao, M.J. and Driscoll, A.R. "The c-chart with bootstrap adjusted control limits to improve conditional  performance", Quality and Reliability Engineering International, 32, pp. 2871-2881 (2016). DOI: 10.1002/qre.1971.
19. Kashif, M., Aslam, M., Rao, G.S., et al. "Bootstrap confidence intervals of the modified process capability index for weibull distribution", Arabian Journal for Science and Engineering, 42, pp. 4565-4573 (2017). DOI: 10.1007/s13369-017-2562-7.
20. Marchant, C., Leiva, V., Cysneiros, F.J.A., et al. "Robust multivariate control charts based on birnbaum- saunders distributions", Journal of Statistical Computation and Simulation, 88(1), pp. 182-202 (2018). DOI: 10.1080/00949655.2017.1381699.
21. Ikpotokin, O. and Siloko, I.U. "A comparative analysis of bootstrap multivariate exponentially-weighted moving average (BMEWMA) control limits", Industrial Engineering & Management Systems, 18(3), pp. 315- 329 (2019). DOI: 10.7232/iems.2019.18.3.315.
22. Mutlu, E.C. and Alakent, B. "Revisiting reweighted robust standard deviation estimators for univariate shewhart S-charts", Quality and Reliability Engineering International, 35, pp. 995-1009 (2019). https://doi.org/10.1002/qre.2441.
23. Koukouvinos, C. and Lappa, A. "A moving average control chart using a robust scale estimator for process dispersion", Quality and Reliability Engineering International, 35, pp. 2462-2493 (2019). DOI: 10.1002/qre.2537.
24. Mahdizadeh, M. and Zamanzade, E. "On interval estimation of the population mean in ranked set sampling", Communications in Statistics-Simulation  and Computation, 51(5), PP. 2747-2768 (2020). https://doi.org/10.1080/03610918.2019.1700276.
25. Mahdizadeh, M. and Zamanzade, E. "Confidence intervals for quantiles in ranked set sampling", Iranian Journal of Science and Technology, Transactions A: Science, 43, pp. 3017-3028 (2019). DOI: 10.1007/s40995-019-00790-6.
26. Ajadi, J.O., Wang, Z., and Zwetsloot, I.M. "A review of dispersion control charts for multivariate individual observations", Quality Engineering, 33(3), pp. 1-16 (2020). DOI: 10.1080/08982112.2020.1755438.
27. Ugaz, W., Alonso, A.M., and Sanchez, I. "Adaptive EWMA-S2 control charts with adaptive smoothing parameter", Quality Engineering, 33(1), pp. 1-13 (2020). DOI: 10.1080/08982112.2020.1776326.
28. Kim, M. and Lee, J. "Geometric charts with bootstrapbased control limits using the bayes estimator", Communications for Statistical Applications and Methods, 27, pp. 65-77 (2020). DOI: 10.29220/CSAM.2020.27.1.065.
29. Moheghi, H.R., Noorossana, R., and Ahmadi, O. "GLM profile monitoring using robust estimators", Quality and Reliability Engineering International, 37(2), pp. 664-680 (2021). DOI: 10.1002/qre.2755.
30. Dizicheh, M.A., Iranpanah, N., and Zamanzade E.  "Bootstrap methods for judgment post stratification", Statistical Papers, 62, pp. 2453-2471 (2021). https://doi.org/10.1007/s00362-020-01197-x.
31. Ahmed, A., Sanaullah, A., and Hanif, M. "A robust alternate to the HEWMA control chart under nonnormality", Quality Technology & Quantitative Management,  17(4), pp. 423-447 (2020). DOI: 10.1080/16843703.2019.1662218.
32. Raza, M.A., Nawaz, T., and Han, D. "On designing distribution-free homogeneously weighted moving average control charts", Journal of Testing and Evaluation, 48(4), pp. 3154-3171 (2020). https://doi.org/10.1520/JTE20180550.
33. Abu-Shawiesh, M.O., Riaz, M., and Khaliq, Q. "MTSD-TCC: A robust alternative to Tukey's control chart (TCC) based on the modified trimmed standard deviation (MTSD)", Mathematics and Statistics, 8(3), pp. 262-277 (2020). DOI: 10.13189/ms.2020.080304.
34. Montgomery, D.C., Introduction to Statistical Quality Control, 6th Edn. Wiley, New York (2011). 
35. Elamir, E. "Probability distribution theory, generalizations and applications of L-moments", Dissertation, Durham University (2001). http://etheses.dur.ac.uk/3987/1/3987 1504.pdf.
36. Saeed, N. and Kamal, S. "New EWMA control charts for monitoring mean under non-normal processes using repetitive sampling", Iranian Journal of Science and Technology, 43(3), pp. 1215-1225 (2018). DOI: 10.1007/s40995-018-0586-9.