Estimators of parameters of a mixture of three multinomial distributions based on simple majority results

For assessing the precision of measurement systems that classify items dichotomically with the possibility of repeated ratings, the maximum likelihood method is commonly used to evaluate misclassification probabilities. However, a computationally simpler and more intuitive approach is the method of simple majority. In this approach, each item is labelled as conforming if the majority of repeated classification outcomes are conforming. A previous study has indicated that this technique yields estimators that have a lower mean squared error than but the same asymptotic properties as the corresponding maximum likelihood estimators. However, there are circumstances in which the use of measurement systems with a wider scale of responses is necessary. In this paper, we propose estimators based on simple majority results for evaluating the classification errors of measurement systems that rate items in a trichotomous domain. We investigate their properties and compare their performance with that of maximum likelihood estimators. We focus on the context in which the true quality states of the objects cannot be determined. The simple majority procedure is modelled using a mixture of three multinomial distributions. The proposed estimators are shown to be a competitive alternative because they offer closed-form expressions and demonstrate a performance similar to that of maximum likelihood estimators.