The Annals of Statistics

Consistency of a recursive estimate of mixing distributions

Surya T. Tokdar, Ryan Martin, and Jayanta K. Ghosh

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Abstract

Mixture models have received considerable attention recently and Newton [Sankhyā 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.

Article information

Source
Ann. Statist., Volume 37, Number 5A (2009), 2502-2522.

Dates
First available in Project Euclid: 15 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1247663763

Digital Object Identifier
doi:10.1214/08-AOS639

Mathematical Reviews number (MathSciNet)
MR2543700

Zentralblatt MATH identifier
1173.62020

Subjects
Primary: 62G07: Density estimation
Secondary: 62G05: Estimation 62L20: Stochastic approximation

Keywords
Mixture models recursive density estimation empirical Bayes

Citation

Tokdar, Surya T.; Martin, Ryan; Ghosh, Jayanta K. Consistency of a recursive estimate of mixing distributions. Ann. Statist. 37 (2009), no. 5A, 2502--2522. doi:10.1214/08-AOS639. https://projecteuclid.org/euclid.aos/1247663763


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