Ryan Martin : Recursive nonparametric estimation of mixing distributions

Posted by , part of the Departmental Colloquium.

At

Feb. 19, 2010, 3 p.m.

In

SEO 636

Abstract

Mixture distributions make for useful statistical models, but estimation of the mixing distribution itself is computationally and theoretically challenging. Recent progress has been made with a fast, online algorithm called predictive recursion (PR), which is capable of producing a continuous estimate of the mixing density. While the computational strengths of PR are readily apparent, good theoretical properties have been much slower to develop. I will present a new and general theorem which says that, if the mixture model is mis-specified, then the PR estimate of the mixture density converges almost surely to the Kullback-Leibler projection of the true density onto the model, provided that the kernel satisfies a certain uniform square-integrability condition. From this, almost sure weak convergence of the PR estimate of the mixing distribution follows as a corollary. The fact that PR is a special case of the Robbins-Monro stochastic approximation process will be used to sketch a proof of the main theorem. I will also give a bound on the rate of convergence, and discuss its minimax nature. Some extensions, applications, and open questions will also be considered.