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April, 1995 Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES
Wing Hung Wong, Xiaotong Shen
Ann. Statist. 23(2): 339-362 (April, 1995). DOI: 10.1214/aos/1176324524


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.


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Wing Hung Wong. Xiaotong Shen. "Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES." Ann. Statist. 23 (2) 339 - 362, April, 1995.


Published: April, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0829.62002
MathSciNet: MR1332570
Digital Object Identifier: 10.1214/aos/1176324524

Primary: 62A10
Secondary: 62F12 , 62G20

Keywords: bracketing metric entropy , Exponential inequality , Hellinger distance , Kullback-Leibler number

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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