Testing for independence in lattice distributions. (Q1856501)

We consider a test for independence between two discrete random variable s with infinite support. It is essentially based on the chi-square statistic with the number of classes diverging to infinity with the sample size \(n\). We prove the asymptotic normality of such test under the null hypothesis as well as its consistency. These results are extended to the class of the so called power-divergence statistics, and a comparison of these tests in terms of Pitman efficiency is given. In order to ensure the results described, a data-based rule of selection of the classes is given.