Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES

Let Y₁,…, Yₙ be independent identically distributed with density p₀ and let F be a space of densities. We show that the supremum of the likelihood ratios ∏ⁿᵢ₌₁ p(Yᵢ)/p₀(Yᵢ), where the supremum is over p ∈ F with \|p1/2 - p1/2₀\|₂ ≥ ε, is exponentially small with probability exponentially close to 1. The exponent is proportional to nε². The only condition required for this to hold is that ε exceeds a value determined by the bracketing Hellinger entropy of F. A similar inequality also holds if we replace F by Fₙ and p₀ by qₙ, where qₙ is an approximation to p₀ 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 εₙ defined by H(εₙ, F) = nε²ₙ, where H(·, F) is the Hellinger entropy of F, is nearly achievable by sieve estimators.