Average Run Length of Cumulative Sum Control Chart by Markov Chain Approach for Zero-Inflated Poisson Processes

Main Article Content

Saowanit Sukparungsee http://orcid.org/0000-0001-5248-8173

Abstract

A Cumulative Sum (CUSUM) chart is an alternatives effective to standard control such Shewhart control chart in order to monitor a small shift in process but limited to normality assumption. Therefore, the objectives in this paper are to propose closed form expression of the Average Run Length (ARL) based on Markov Chain Approach (MCA) underlying Zero-Inflated Poisson (ZIP) processes. Furthermore, performance of the CUSUM chart is compared with an Exponentially Weighted Moving Average (EWMA) chart by studying the effect of the probability of extra zeros in ZIP models and the magnitudes of shifts in parameter of the ZIP process for both of the control charts. The numerical results are obtained from the MCA which the performance of EWMA and CUSUM charts are in the same manner for the smallest magnitude of shifts  However, the CUSUM shows the poor performance for moderate to large shifts when compared with the EWMA chart. 

Downloads

Download data is not yet available.

Article Details

Section
Articles
Author Biography

Saowanit Sukparungsee, King Mongkut's University of Technology, North Bangkok

Department of Applied Statistics,

Faculty of Applied Sciences

References

Areepong Y and Novikov A. Martingale approach to EWMA control chart for changes in exponential distribution. Journal of Quality Measurement and Analysis. 2008; 4: 197-203

Areepong Y and Sukparungsee S. Explicit expression for the average run length of double moving average scheme for zero-inflated binomial process. International Journal of Applied Mathematics and Statistics. 2015; 53(3): 33-43

Areepong Y and Sukparungsee S. Approximation average run lengths of zero-inflated binomial GWMA chart with Markov chain approach. Far East Journal of Mathematical Sciences. 2016; 99(3): 413-428

B̈ohning D, Dierz E and Schlattmann P. The zero-inflated Poisson model and the decayed missing and filled teeth index in dental epidemiology, J R Stat Soc Ser A Stat Soc. 1999; 162(2), 195-209

Broek J. A score test for zero inflation in a Poisson distribution. Biometrics. 1995; 51: 738–743

Brook D and Evans DA. An approach to the probability distribution of CUSUM run length. Biometrika. 1972; 9: 539-548

Busaba J, Sukparungsee S, Areepong Y and Mititelu G. An analysis of average run length for CUSUM procedure with negative exponential data. Chiang Mai Journal of Science. 2012; 39(2): 200-208

Chananet C, Areepong Y and Sukparungsee S. A markov chain approach for average run length of EWMA and CUSUM control chart based on ZINB Model. International Journal of Applied Mathematics and Statistics. 2015; 53(1): 126-137

Chananet C, Areepong Y and Sukparungsee S. An approximate formula for ARL in moving average chart with ZINB data. Thailand Statistician. 2015; 13(2): 209-222

Cohen AC. Estimation in Mixtures of Discrete Distributions. Proceedings of the International Symposium on Discrete Distributions, Montreal. 1963; 373-378

Crowder SV. A simple method for studying run length distributions of exponentially weighted moving average charts. Technometrics. 1978; 29: 401-407

Fatahi AA, Noorossana R, Dokouhaki P and Moghaddam BF. Zero inflated Poisson EWMA contrl chart for monitoring rare health-related events. Published on line, J Mech Med Biol. 2012; 12(4): DOI: 10.1142/S0219519412500650

Lambert D. Zero Inflated Poisson Regression with and application to defects in manufacturing. Technometrics. 1992; 34: 1-14

Lucas JM and Saccucci MS. Exponentially weighted moving average control schemes: properties and enhancements. Technometrics. 1990; 32: 1-29

Montgomery DC. Introduction to statistical quality control. New York: Wiley; 2009

Petcharat K, Areepong Y, Sukparungsee S and Mititelu G. Exact solution for average run length of CUSUM charts for MA(1) process. Chiang Mai Journal of Science. 2014; 41(5.2): 1449-1456

Petcharat K, Sukparungsee S and Areepong Y. Exact solution of the average run length for the cumulative sum charts for a moving average process of order q. Science Asia. 2015; 41: 141-147

Roberts SW. Control chart tests based on geometric moving average. Technometrics. 1959; 1: 239-250

Somran S, Sukparungsee S and Areepong Y. Analytic and numerical solutions of ARL of CUSUM procedure for exponentially distributed observations. Thailand Statistician. 2016; 14(1): 249-258

Sukparungsee S and Novikov AA. Analytical approximations for detection of a change-point in case of light-tailed distributions. Journal of Quality Measurement and Analysis. 2008; 4: 373-380

Suriyakat W, Areepong Y, Sukparungsee S and Mititelu G. An analytical approach to EWMA control chart for AR(1) process observations with exponential white noise. Thailand Statistician. 2012; 10(1): 41-52

Unhapipat S. Inference for zero-inflated Poisson distribution. PhD [dissertation]. Mahidol University; 2015

Woodall WH and Adams BM. The statistical design of CUSUM charts. Qual Eng. 1993; 4: 559–570