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Consistency of a recursive estimate of mixing distributions. (English) Zbl 1173.62020
Summary: Mixture models have received considerable attention recently and M. A. Newton [Sankhyā, Ser. A 64, 306–322 (2002)] 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.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62L12 Sequential estimation
65C60 Computational problems in statistics (MSC2010)
65C05 Monte Carlo methods
62G05 Nonparametric estimation
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