Refining the Convergence Rate for $\phi$-Divergence Statistics in Multinomial Goodness-of-Fit Tests

Abstract

We study asymptotic properties of statistics in goodness-of-fit tests under the simple null hypothesis of a multinomial distribution. It is also assumed that the statistics are based on $\phi$-divergences, the sample size $n$ tends to infinity, and the number of cells in the group $k$ is fixed. It is known that in this case the limiting law is the chi-square distribution, and according to the classical theory, the distribution function of the test statistic converges to the limiting one at rate $O(n^{-1/2})$. A distinctive feature of the present paper is the use of an approach in which the original problem is reduced to a well-known problem in number theory, namely, to the generalized Gauss problem on the number of “integer” points in an expanding convex set with smooth boundary. This makes it possible to refine the order of convergence to $ O(n^{-1+\alpha(k)})$, where $\alpha(k) > 0$ and $\alpha(k)$ decreases to zero with increasing $k$. Thus, a result previously known only for the family of power divergence statistics is extended to a wider class of statistics based on $\phi$-divergences.