Statistical inference in constrained latent class models for multinomial data based on $$\phi $$ ϕ -divergence measures

Abstract

In this paper we explore the possibilities of applying $$\phi $$ -divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum $$\phi $$ -divergence estimators (M $$\phi $$ Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M $$\phi $$ Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M $$\phi $$ Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, $$\phi $$ -divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, $$\phi $$ -divergence measures can handle the problem and we derive a family of $$\phi $$ -divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.