Stochastic Approximation and Newton's Estimate of a Mixing Distribution

Details Stochastic Approximation and Newton's Estimate of a Mixing Distribution Stochastic Approximation and Newton's Estimate of a Mixing Distribution

2011-02-17

arxiv.org/abs/1102.3592v1

Statistical Science 2008, Vol. 23, No. 3, 365-382

Statistics

Methodology

Published in at http://dx.doi.org/10.1214/08-STS265 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M

Scientific paper

10.1214/08-STS265

Many statistical problems involve mixture models and the need for computationally efficient methods to estimate the mixing distribution has increased dramatically in recent years. Newton [Sankhya Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating the mixing distribution, which we study as a special case of stochastic approximation (SA). We begin with a review of SA, some recent statistical applications, and the theory necessary for analysis of a SA algorithm, which includes Lyapunov functions and ODE stability theory. Then standard SA results are used to prove consistency of Newton's estimate in the case of a finite mixture. We also propose a modification of Newton's algorithm that allows for estimation of an additional unknown parameter in the model, and prove its consistency.

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