  Predictive recursion (PR) is a fast stochastic algorithm for nonparametric
estimation of mixing distributions in mixture models. It is known that the PR
estimates of both the mixing and mixture densities are consistent under fairly
mild conditions, but currently very little is known about the rate of
convergence. Here I first investigate asymptotic convergence properties of the
PR estimate under model misspecification in the special case of finite mixtures
with known support. Tools from stochastic approximation theory are used to
prove that the PR estimates converge, to the best Kullback--Leibler
approximation, at a nearly root-$n$ rate. When the support is unknown, PR can
be used to construct an objective function which, when optimized, yields an
estimate the support. I apply the known-support results to derive a rate of
convergence for this modified PR estimate in the unknown support case, which
compares favorably to known optimal rates.
