The probabilities of large deviations for a certain class of statistics associated with multinomial distribution

Abstract

Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems.