  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.
