Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES

Abstract

Let $Y_1,\ldots, Y_n$ be independent identically distributed with density $p_0$ and let $\mathscr{F}$ be a space of densities. We show that the supremum of the likelihood ratios $\prod^n_{i=1} p(Y_i)/p_0(Y_i)$, where the supremum is over $p \in \mathscr{F}$ with $\|p^{1/2} - p^{1/2}_0\|_2 \geq \varepsilon$, is exponentially small with probability exponentially close to 1. The exponent is proportional to $n\varepsilon^2$. The only condition required for this to hold is that $\varepsilon$ exceeds a value determined by the bracketing Hellinger entropy of $\mathscr{F}$. A similar inequality also holds if we replace $\mathscr{F}$ by $\mathscr{F}_n$ and $p_0$ by $q_n$, where $q_n$ is an approximation to $p_0$ in a suitable sense. These results are applied to establish rates of convergence of sieve MLEs. Furthermore, weak conditions are given under which the "optimal" rate $\varepsilon_n$ defined by $H(\varepsilon_n, \mathscr{F}) = n\varepsilon^2_n$, where $H(\cdot, \mathscr{F})$ is the Hellinger entropy of $\mathscr{F}$, is nearly achievable by sieve estimators.