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Estimating the mixing density of a mixture distribution remains aninteresting problem in statistics literature. Using a stochastic approximationmethod, Newton and Zhang (1999) introduced a fast recursive algorithm forestimating the mixing density of a mixture. Under suitably chosen weights thestochastic approximation estimator converges to the true solution. In Tokdaret. al. (2009) the consistency of this recursive estimation method wasestablished. However, the proof of consistency of the resulting estimator usedindependence among observations as an assumption. Here, we extend theinvestigation of performance of Newton's algorithm to several dependentscenarios. We prove that the original algorithm under certain conditionsremains consistent even when the observations are arising from a weaklydependent stationary process with the target mixture as the marginal density.We show consistency under a decay condition on the dependence amongobservations when the dependence is characterized by a quantity similar tomutual information between the observations.

Stochastic Approximation Algorithm for Estimating Mixing Distribution for Dependent Observations