A control chart using copula-based Markov chain models

Long, Ting-Hsuan and Emura, Takeshi (2014): A control chart using copula-based Markov chain models. Forthcoming in: Journal of the Chinese Statistical Association

Abstract

Statistical process control is an important and convenient tool to stabilize the quality of manufactured goods and service operations. The traditional Shewhart control chart has been used extensively for process control, which is valid under the independence assumption of consecutive observations. In real world applications, there are many types of dependent observations in which the traditional control chart cannot be used. In this paper, we propose to apply a copula-based Markov chain to perform statistical process control for correlated observations. In particular, we consider three methods to obtain the estimates of upper control limit (UCL) and lower control limit (LCL) for the control chart. It is shown by simulations that Joe’s parametric maximum likelihood method provides the most reliable estimates of the UCL and LCL compared to the other methods. We also propose simulation techniques to compute the average run length (ARL) of the proposed charts, which can be used to set the UCL and LCL for a given value of ARL. The piston rings data are analyzed for illustration.

• Item Type: MPRA Paper
• Language: English
• Keywords: Average run length, Clayton model, correlated data, Kendall’s tau, Markov chain
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General

References

Bagshow, M., Johnson, R. A. (1975). The effect of serial correlation on the performance of CUSUM tests II. Technometrics 17, 73-80.

Box, G., Narasimhan, S. (2010), Rethinking statistics for quality control. Quality Engineering 22, 60-72.

Chen, X., Fan, Y. (2006). Estimation of copula-based semiparametric time series models. Journal of Econometrics 130, 307-335.

Chen, C. C., Fuh, C. D. and Teng, H. W. (2013). Efficient option pricing with importance sampling. Journal of the Chinese Statistical Association 51, 253–273.

Darsow, W. F., Nguten B., Olsen, E. T. (1992). Copulas and Markov Processes. Illinois Journal of Mathematics 36, 600-642.

Emura, T., Konno, Y. (2012). A goodness-of-fit tests for parametric models based on dependently truncated data. Computational Statistics & Data Analysis 56, 2237-2250.

Emura, T. and Chen, Y.H. (2014), Gene selection for survival data under dependent censoring: a copula-based approach, Statistical Methods in Medical Research, doi: 10.1177/0962280214533378.

Emura T., Lin Y. S. (2013), A comparison of normal approximation rules for attribute control charts. Quality and Reliability Engineering International, doi: 10.1002/qre.1601.

Frees, E. W., Valdez, E. (1998). Understanding the relationships using copulas. North American Actuarial Journal 2, 1-25.

Genest, C., Rémillard, B. (2008). Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Annales de Institut Henri Poicare -Probabilites et Statistiques 44, 1096-1127.

Genest, C., Nešlehová, J.G., Rémillard, B. (2013). On the estimation of Spearman’s rho and related tests of independence for possibly discontinuous multivariate data. Journal of Multivariate Analysis 117, 217-228.

Hung, Y. C., Tseng, N. F. 2013. Extracting informative variables in the validation of two-group causal relationship. Computational Statistics 28, 1151-1167.

Hu, Y. H. (2014). Maximum likelihood estimation for double-truncation data under a special exponential family, Master Thesis, Graduate Institute of Statistics, National Central University, Taiwan.

Hryniewicz, O. (2012). On the robustness of the Shewhart control chart to different types of dependencies in data. Frontiers in Statistical Quality Control 10, Lenz, H.-J. et al. (Eds.), Springer-Verlag Berlin Heidelberg.

Joe, H. (1997). Multivariate Models and Dependence Concepts. CHAPMAN & HALL/CRC.

Johnson, R. A., Bagshaw, M. (1974). The effect of serial correlation on the performance of CUSUM tests. Technometrics 16, 103-112.

Knight, K. (2000). Mathematical Statistics. Chapman & Hall.

Knoth, S., Schmid, W. (2004). Control charts for time series: a review. Frontiers in Statistical Quality Control 7, Lenz, H.-J. et al. (Eds.), Springer-Verlag Berlin Heidelberg.

Kramer, H. G., Schmid, W. (2000). The influence of parameter estimation on the ARL of Shewhart type charts for time series. Statistical Papers 41, 173-196.

Luca L. Quality control charts, R qcc package, version 2.4. 2014.

Montgomery, D. C. (2009a). Statistical Quality Control, Sixth Edition. Wiley.

Montgomery, D. C. (2009b). Introduction to Statistical Quality Control, Sixth Edition. Wiley.

Nelsen, R. B. (2006). An Introduction to Copulas, 2nd Edition. Springer Series in Statistics, Springer-Verlag. New York.

Nešlehová, J. (2007). On rank correlation measures for non-continuous random variables, Journal of Multivariate Analysis 98, 544-567.

Psarakis, S., Papaleonida, G. E. A. (2007). SPC procedures for monitoring autocorrelated processes, Quality Techinology & Quantitative Management 4 (4), 501-540.

Ross, S. M. (2013). Simulation, Fifth Edition. Elsevier.

Sari, J. K., Newby, M. J., Brombacher, A. C., and Tang, L. C. (2009), Bivariate constant stress degradation model: led lighting system reliability estimation with two-stage monitoring, Quality and Reliability Engineering International 25, 1067-1084.

Schmid, W. (1995). On the run length of a Shewhart chart for correlated data. Statistical Papers 36, 111-130.

Sklar, A. (1959). Fonctions de re'partition a' n dimensions et leurs marges. Publications de l'Intitut de Statistique de l'Universit de Paris 8, 229-231.

Vasilopoulos, A. V., Stamboulis, A. P. (1978), Modification of control chart limits in the presence of data correlation. J. Quality Technology 10 (1), 20-30.

Wardell, D. G., Moskowitz, H, Plante, R. D. (1994). Run-length distributions of special-cause control charts for correlated process. Technometrics 36, 3-27.

Wetherill G. B., Brown D. W. (1991) Statistical process control, theory and practice. Chapman and Hall.

Wieringa, J. E. (1999) Statistical process control for serially correlated data, Labyrint Publishing.