Rates of convergence for Bayes and maximum likelihood estimation for mixture of normal densities

S. Ghosal, A.W. van der Vaart

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Abstract

We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or location-scale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub-Gaussian tails. We obtain bounds for Hellinger bracketing entropies for this class, and from these bounds, we deduce the convergence rates of (sieve) MLEs in Hellinger distance. The rate turns out to be (log n)
Original languageEnglish
Pages (from-to)1233-1263
Number of pages31
JournalAnnals of Statistics
Volume29
Issue number5
DOIs
Publication statusPublished - 2001

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