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

Sina Molavipour; Germán Bassi; Mikael Skoglund

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arXiv · 1808.05771

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

Sina Molavipour·Germán Bassi·Mikael Skoglund

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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.

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Sina Molavipour, Germán Bassi, Mikael Skoglund. 2018-08-17. Non-Asymptotic Behavior of the Maximum Likelihood Estimate of a Discrete Distribution. https://arxiv.org/abs/1808.05771

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