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

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.