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

Institute of Mathematics, Academy of Sciences of Uzbekistan, Mirzo Ulugbek str., 81, 100125 Tashkent, Uzbekistan.

^* Corresponding author: shmirakhmedov@yahoo.com

Received: 8 October 2019
Accepted: 3 August 2020

Abstract

Let η = (η₁, …, η_N) be a multinomial random vector with parameters n = η₁ + ⋯ + η_N and p_m > 0, m = 1, …, N, p₁ + ⋯ + p_N = 1. We assume that N →∞ and maxp_m → 0 as n →∞. The probabilities of large deviations for statistics of the form h₁(η₁) + ⋯ + h_N(η_N) are studied, where h_m(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.