University of Washington - Department of Mathematics
This lecture focuses on problems of density estimation (both parametric and nonparametric) and if there is time, time series estimation (no pun intended). When formulated as optimization problems, consistency of the estimators becomes a question of whether a sequence of optimization problems converge in an appropriate sense to the true problem. The tools of variational analysis are used to examine the question of consistency for these problems. In particular, an epigraphical ergodic theorem can be used to show consistency for a broad class of estimation problems. One consequence of this approach is the realization that one may incorporate prior information (e.g. monotonicity, smoothness) in the form of constraints in an optimization problem (and still retain consistency). This has the potential to lead to much better numerical results when the sample size is small or the sample space is of high dimension (e.g. greater than or equal to 2).