Gestão & Produção
Gestão & Produção
Seção Temática: Monitoramento e Controle Estatístico de Processos

Linear combination of chi-squares for multinomial process monitoring

Ramzi Talmoudi; Ali Achouri; Hassen Taleb

Downloads: 0
Views: 68


Abstract:: Marcucci (1985) proposed a chi square goodness of fit statistic based generalized p-chart for multinomial process monitoring. A chi square distribution quantile was considered as a control chart limit. A weighted chi square goodness of fit statistic-based control chart is proposed for multinomial process monitoring in this paper, where more important weights are advocated to poor quality categories. The statistic distribution is approximated by a well-known linear combination of chi squares distribution. The approximation is assessed through a simulation, an extreme percentile of the approximated distribution is used as an upper control chart limit and a comparison is carried out with a chi square goodness of fit statistic-based control chart. The average run length is used as a benchmark and the comparison is performed using simulations considering two process shifts scenarios. Under some restrictions, the weighted statistic-based control chart allows an earlier detection of process shift in case of deterioration and postpones out of control signals in case of improvement. This benefit is clearer when the process is improved by a decrease in the poor quality probability category and an increase in the best quality category probability.


Multinomial Process, Generalized p-Chart, Distribution Approximation, Simulation


Bodenham, D. A., & Adams, N. M. (2016). A comparison of efficient approximations for a weighted sum of chi-squared random variables. Statistics and Computing, 26(4), 917-928.

Castano-Martinez, A., & Lopez-Blazquez, F. (2005). Distribution of a sum of weighted noncentral chi-square variables. Test, 14(2), 397-415.

Cozzucoli, P. C. (2009). Process monitoring with multivariate p-Control chart. International Journal of Quality, Statistics, and Reliability, 2009, 1-11.

Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of chi-square random variables. Journal of the Royal Statistical Society. Series C, Applied Statistics, 29(3), 323-333.

Davis, A. W. (1977). A differential equation approach to linear combination of independent chi-squares. Journal of the American Statistical Association, 72(357), 212-214.

Duncan, A. J. (1950). A Chi-Square chart for controlling a set of percentages. Industrial Quality Control., 7, 11-15.

Feiveson, A. H., & Delaney, F. C. (1968). The distribution and properties of a weighted sum of chi squares (NASA Technical Note, D-4575). Houston, Texas: Manned Spacecraft Center.

Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48(3-4), 419-426.

Jensen, D. R., & Solomon, H. (1972). A Gaussian approximation to the distribution of a definite quadratic form. Journal of the American Statistical Association, 67(340), 898-902.

Kotz, S., Johnson, N. L., & Boyd, D. W. (1967). Series representations of distributions of quadratic forms in normal variables. I. Central case. Annals of Mathematical Statistics, 38(3), 823-837.

Li, J., Tsung, F., & Zou, C. (2014a). A simple categorical chart for detecting location shifts with ordinal information. International Journal of Production Research, 52(2), 550-562.

Li, J., Tsung, F., & Zou, C. (2014b). Multivariate binomial/multinomial control chart. IIE Transactions, 46(5), 526-542.

Marcucci, M. (1985). Monitoring multinomial processes. Journal of Quality Technology, 17(2), 86-91.

Mathai, A. M., & Provost, S. B. (1992). Quadratic forms in random variables: Theory and Applications (vol. 126). New York: Marcel Dekker Inc.

Moschopoulos, P. G., & Canada, W. B. (1984). The distribution function of linear combination of chi-squares. Computers & Mathematics with Applications (Oxford, England), 10(4/5), 383-386.

Oman, S. D., & Zacks, S. (1981). A mixture approximation to the distribution of a weighted sum of chi-squared variables. Journal of Statistical Computation and Simulation, 13(3-4), 215-224.

Perry, M. B. (2019). An EWMA control chart for categorical processes with applications to social network monitoring. Journal of Quality Technology.

Raz, T., & Wang, J. (1990). On the construction of control charts using linguistic variables. International Journal of Production Research, 28(3), 477-487.

Taleb, H., & Limam, M. (2002). On fuzzy and probabilistic control charts. International Journal of Production Research, 40(12), 2849-2863.

Taleb, H., Limam, M., & Hirota, H. (2006). Multivariate fuzzy multinomial control charts. Quality Technology & Quantitative Management, 3(4), 437-453.

Topalidou, E., & Psarakis, S. (2009). Review of multinomial and multi-attribute quality control charts. Quality and Reliability Engineering International, 25(7), 773-804.

Weiss, C. H. (2018). Control Charts for Time-Dependent Categorical Processes. In: S. Knoth & W. Schmid (Eds.) Frontiers in statistical quality control 12 (pp. 211-231). Springer International Publishing.

Yashchin, E. (2012). On detection of changes in categorical data. Quality Technology & Quantitative Management, 9(1), 79-96.

6138079aa953955ea3485973 gp Articles

Gest. Prod.

Share this page
Page Sections