A Posterior p-Value for Homogeneity Testing of the Three Sample Problem

In this paper we study a special kind of finite mixture model. The sample drawn from the model consisits of three parts. The first two parts are drawn from specified distributions f1 and f2, while the third one is drawn from the mixture. A problem of interest is whether the two distributions f1 and f2 are the same. To test this hypothesis, we first define the regular location and scale family of distributions and assume that f1 and f2 are regular denisty functions. Then the hypothesis transforms to the equalities of the location and scale parameters, respectively. To utilize the information in the sample, we use Bayes’ theorem to obtain the posterior distribution and give the sampling method. Then we propose the posterior p-value to test the hypothesis. The simulation studies show that our posterior p-value largely improves the power in both normal and logistic cases while nicely controls the Type-I error. A real halibut dataset is used to illustrate the validity of our method.