Phi-divergence Test Statistics in Multinomial Sampling for Hierarchical Sequences of Loglinear Models with Linear Constraints

We consider nested sequences of hierarchical loglinear models when expected frequencies are subject to linear constraints and we study the problem of finding the model in the the nested sequence that is able to explain more clearly the given data. It will be necessary to give a method to estimate the parameters of the loglinear models and also a procedure to choose the best model among the models considered in the nested sequence under study. These two problems will be solved using the φ -divergence measures. We estimate the unknown parameters using the minimum φ -divergence estimator (Martín and Pardo [8]) which can be considered as a generalization of the maximum likelihood estimator (Haber and Brown [5]) and we consider a φ -divergence test statistic (Martín [7]) that generalize the likelihood ratio test as well as the chi-square test statistic, for testing two nested loglinear models.