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

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.

Nonparametric Versus Parametric Reasoning Based on Contingency Tables

This paper proposes scenarios of generating two-way and three way contingency tables (CTs). A concept of probability flow parameter (PFP) plays a crucial role in these scenarios. Additionally, measures of untruthfulness of H0 are defined. The power divergence statistics and the |X| statistics are used. This paper is a simple attempt to replace a nonparametric statistical inference from CTs by the parametric one. Maximum likelihood method is applied to estimate PFP and instructions of generating CTs according to scenarios in question are presented. The Monte Carlo method is used to carry out computer simulations.

A New Test for Independence in 2×2 Contingency Tables

In statistical literature there exist many tests to reveal the independence of two qualitative variables in two‑way contingency tables (CTs), in particular in 2×2 CTs. In this paper four independence tests were compared. These are: the chi‑square test, being the most popular type of power divergence statistics; the modular test and the d‑square test, which is a modification of the Pearson’s test; the logarithmic minimum test which is a new proposal. Critical values for the tests listed above were determined with the Monte Carlo method. In order to compare the tests, the measure of untruthfulness of H0 was proposed and the power of the tests was calculated.