  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.
