An empirical bayes process monitoring technique for polytomous data

When a product item is tested, usually one has more information than just pass or fail. Often there are categories of failure modes. The purpose of this paper is to develop a method to monitor the fractions of the tested items falling into different categories of pass/fail modes. Using the multinomial model with Dirichlet prior, we describe the theory underlying an empirical Bayes approach to monitoring polytomous data generated in manufacturing processes. A pseudo maximum likelihood estimator (PMLE) and the method-of-moments estimator (MME) of the hyperparameters of the prior distribution are considered and compared by a simulation study. It is found that the PMLE performs slightly better than the MME. A monitoring scheme based on the marginal distributions of the observed pass/fail fractions is proposed. The average run length behavior of the proposed monitoring scheme is investigated. Finally, an example to illustrate the use of the technique is given.