Linear combination of chi-squares for multinomial process monitoring

Abstract

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.