  We study the relative entropy of the empirical probability vector with
respect to the true probability vector in multinomial sampling of $k$
categories, which, when multiplied by sample size $n$, is also the
log-likelihood ratio statistic. We generalize a recent result and show that the
moment generating function of the statistic is bounded by a polynomial of
degree $n$ on the unit interval, uniformly over all true probability vectors.
We characterize the family of polynomials indexed by $(k,n)$ and obtain
explicit formulae. Consequently, we develop Chernoff-type tail bounds,
including a closed-form version from a large sample expansion of the bound
minimizer. Our bound dominates the classic method-of-types bound and is
competitive with the state of the art. We demonstrate with an application to
estimating the proportion of unseen butterflies.
