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.


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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