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Journal Article

# Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES

Wing Hung Wong and Xiaotong Shen
The Annals of Statistics
Vol. 23, No. 2 (Apr., 1995), pp. 339-362
Stable URL: http://www.jstor.org/stable/2242340
Page Count: 24

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## Abstract

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 \$\prod^n_{i=1} p(Y_i)/p_0(Y_i)\$, where the supremum is over p ∈ F with |p1/2 - p1/2 0|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) = nε2 n, where H(·, F) is the Hellinger entropy of F, is nearly achievable by sieve estimators.

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