Consistency of a recursive estimate of mixing distributions

TL;DR

This paper proves the almost sure consistency of a recursive estimator for mixing distributions in mixture models, introduces a permutation-invariant modification, and compares its finite-sample performance with other estimators.

Contribution

It establishes the consistency of Newton's recursive estimator and proposes a permutation-invariant version, enhancing its theoretical robustness.

Findings

Recursive estimate is consistent under mild conditions.

Permutation-invariant modification improves estimator stability.

Simulation results compare performance of various estimators.

Abstract

Mixture models have received considerable attention recently and Newton [Sankhy\={a} 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.