A Note for Likelihood Ratio Methods for Testing the Homogeneity of a Three-Sample Problem with a Mixture Structure

Recently, our paper entitled “Generalized fiducial methods for testing the homogeneity of a three-sample problem with a mixture structure” is published in Journal of Applied Statistics 50 (2023), pp. 1094–1114. In simulation studies of this paper, the likelihood ratio method is regarded as a comparison method with the generalized fiducial methods. However, the construction of the likelihood ratio method and its asymptotic theories were not provided. It is worth noting that under the null model, the proportion parameter disappears, and it is unidentifiable. Hence, the classic theory of the likelihood ratio method is not applicable to the testing problem we consider. In consideration of the advantages owned by the likelihood ratio method, it is necessary to separately to establish the likelihood ratio method and study its asymptotic theory. Therefore, we write this note to highlight this method.