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Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. (English) Zbl 0829.62002
Summary: Let $$Y_1, \dots, Y_n$$ be independent, identically distributed with density $$p_0$$ and let $$\mathcal F$$ 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 $$p \in {\mathcal F}$$ 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 $$\mathcal F$$. A similar inequality also holds if we replace $$\mathcal F$$ by $${\mathcal F}_n$$ 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 $$H(\varepsilon_n, {\mathcal F}) = n \varepsilon^2_n$$, where $$H(\cdot, {\mathcal F})$$ is the Hellinger entropy of $$\mathcal F$$, is nearly achievable by sieve estimators.

##### MSC:
 62A01 Foundations and philosophical topics in statistics 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 60E15 Inequalities; stochastic orderings
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