Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. (English) Zbl 0829.62002

Wong, Wing Hung; Shen, Xiaotong

Ann. Stat. 23, No. 2, 339-362 (1995).

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

Keywords:

Hellinger distance; bracketing metric entropy; Kullback-Leibler number; exponential inequality; truncation; space of densities; supremum; likelihood ratios; Hellinger entropy; rates of convergence; sieve estimators