Non-Asymptotic Behavior of the Maximum Likelihood Estimate of a Discrete Distribution

TL;DR

This paper provides explicit finite-sample bounds for the distribution of the log-likelihood ratio in discrete distributions, quantifying its convergence to the chi-squared distribution as sample size increases.

Contribution

It derives a non-asymptotic bound on the difference between the log-likelihood ratio's distribution and the chi-squared distribution, extending Wilks' theorem to finite samples.

Findings

Explicit bound on the cdf difference of the likelihood ratio and chi-squared distribution.

The bound decreases at a rate of 1/√n, confirming Wilks' theorem in finite samples.

Quantitative characterization of the non-asymptotic behavior of MLE for discrete distributions.

Abstract

In this paper, we study the maximum likelihood estimate of the probability mass function (pmf) of $n$ independent and identically distributed (i.i.d.) random variables, in the non-asymptotic regime. We are interested in characterizing the Neyman--Pearson criterion, i.e., the log-likelihood ratio for testing a true hypothesis within a larger hypothesis. Wilks' theorem states that this ratio behaves like a $\chi^2$ random variable in the asymptotic case; however, less is known about the precise behavior of the ratio when the number of samples is finite. In this work, we find an explicit bound for the difference between the cumulative distribution function (cdf) of the log-likelihood ratio and the cdf of a $\chi^2$ random variable. Furthermore, we show that this difference vanishes with a rate of order $1/\sqrt{n}$ in accordance with Wilks' theorem.