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Stochastic Approximation and Newton's Estimate of a Mixing Distribution

By Ryan Martin and Jayanta K. Ghosh


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.Comment: Published in at the Statistical Science ( by the Institute of Mathematical Statistics (

Topics: Statistics - Methodology
Year: 2011
DOI identifier: 10.1214/08-STS265
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