Journal
ANNALS OF STATISTICS
Volume 37, Issue 5A, Pages 2502-2522Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/08-AOS639
Keywords
Mixture models; recursive density estimation; empirical Bayes
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Mixture models have received considerable attention recently and Newton [Sankhya Ser A 64 (2002) 306-322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao-Blackwell argument is used to prove consistency in probability of this alternative estimate. Several Simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate.
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