= Compact-parameter consistency of maximum likelihood
{title2=$\widehat\theta_n\xrightarrow{p}\theta_0$}
On a compact parameter space, suppose the expected log-likelihood is continuous, uniquely maximized at the true parameter, and the normalized empirical log-likelihood converges uniformly in probability to its expectation. On the compact complement of every neighbourhood of the truth, the expected criterion has a strictly positive gap below its maximum. Uniform approximation by less than one-third of this gap excludes every empirical maximizer from that complement. <Identifiability> and nonnegativity of <Kullback-Leibler divergence> supply the unique expected maximizer. The proof works for any measurable choice of maximizer.
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