The demerit‐based control chart for trinomial distribution

Purpose – The purpose of this paper is to investigate the properties of the classical goodness of fit test statistics X 2 , G 2 , GM 2 , and NM 2 in testing quality of process represented as the trinomial distribution with dip null hypothesis and to devise a control chart for the trinomial distribution with dip null hypothesis based on demerit control chart. Design/methodology/approach – The research involves the linear form of the test statistics, the linear function of counts since the marginal distribution of the counts in any category is binomial or approximated Poisson, in which the uniformly minimum variance unbiased estimator is the linear function of counts. A control chart is used for monitoring student characteristics in Thailand. The control chart statistics based on an average of the demerit value computed for each student as a weighted average lead to a uniformly most powerful unbiased test marginally. The two‐sided control limits were obtained using percentile estimates of the empirical distribution of the averages of the demerit. Findings – The demerit control chart of the weight set (1, 25, 50) shows a generally good performance, robust to direction of out‐of‐control, mostly outperforms the GM 2 and is recommended. The X 2 , NM 2 are not recommended in view of inconsistency and bias. The performance of the demerit control chart of the weight set (1, 25, 50) does not dramatically change between both directions. Practical implications – None of the multivariate control charts for counts presented in the literature deals with trinomial distribution representing the practical index of the quality of the production/process in which the classification of production outputs into three categories of “good”, “defective”, and “reworked” is common. The demerit‐based control chart presented here can be applied directly to this situation. Originality/value – The research considers how to deal with the trinomial distribution with dip null hypothesis which no research study so far has presented. The study shows that the classical Pearson's X 2 , Loglikelihood, modified Loglikelihood, and Neyman modified X 2 could fail to detect an “out‐of‐control”. This research provides an alternative control chart methodology based on demerit value with recommended weight set (1, 25, 50) for use in general.