Chi-squared goodness-of-fit tests: The optimal choice of grouping intervals

Abstract

When using the chi-squared goodness-of-fit tests, the problem of choosing boundary points and the number of grouping intervals is always urgent, as the power of these tests considerably depends on the grouping method used. In this paper, the investigation of the power of the Pearson and Nikulin-Rao-Robson chi-squared tests has been carried out for various numbers of intervals and grouping methods. The partition of the real line into equiprobable intervals is not an optimal grouping method, as a rule. It has been shown that asymptotically optimal grouping, for which the loss of the Fisher information from grouping is minimized, enables to maximize the power of the Pearson test against close competing hypotheses. In order to find the asymptotically optimal boundary points, it is possible to maximize some functional (the determinant, the trace or the minimum eigenvalue) of the Fisher information matrix for grouped data. The versions of asymptotically optimal grouping method maximize the test power relative to a set of close competing hypotheses, but they do not insure the largest power against some given competing hypothesis. For the given competing hypothesis H1, it is possible to construct the chi-squared test, which has the largest power for testing hypothesis H0 against H1. For example, in the case of the Pearson chi-squared test, it is possible to maximize the non-centrality parameter for the given number of intervals. So, the purpose of this paper is to give the methods for the choice of optimal grouping intervals for chi-squared goodness-of-fit tests.