The Behavior of Chi-Square Statistics Using Unbiased Estimators in the Case of One-Parameter Distribution from Exponential Family

Abstract

We propose two new variants of chi-square-type statistics for testing the hypothesis on the shape of a distribution belonging to the one-parameter exponential family. Each variant assumes the use of unbiased estimators of the probabilities for the random variable to hit the atoms (intervals) of decomposition of its range. We establish the convergence of each statistics’ distribution to the chi-square distribution with (r − 1) degrees of freedom, where r is the number of decomposition atoms.