Spring 2002 Seminar Series - February 28, 2002
University of Minnesota
School of Statistics
College of Liberal Arts

Exact Minimax Density Estimation

Feng Liang
Yale University
(Statistics Search Candidate)

Thursday, February 28, 2002
4:00 PM, B10 Ford Hall
Minneapolis, East Bank Campus
Social at 3:30 PM, 300 Ford Hall

Abstract

For problems of model selection in regression, we determine an exact minimax universal data compression strategy for the minimum description length (MDL) criterion. The analysis also gives the best invariant and indeed minimax procedure for predictive density estimation in location families and scale families, using Kullback-Leibler loss. The exact minimax rule is a generalized Bayes using a uniform (Lebesgue measure) prior on the location parameter for location families and on the log-scale for the scale families. Such improper priors are made proper by conditioning on an initial set of observations.
Our proof for the minimaxity already implies the admissibility for location families in one dimension. However, it is well known that the best invariant estimator might not be admissible. For example, for normal location families, the sample mean is not admissible when dimension is three or higher (Stein, 55). Moreover, there exists a proper Bayes estimator which produces improved risk everywhere than the sample mean (Strawderman, 71), when dimension is bigger than four. We present an analogous result for predictive density estimation, using Kullback-Leibler loss.