University of Washington - Department of Statistics
Empirical process methods play an important role in the study of maximum likelihood and minimum contrast estimators in non-parametric and semi-parametric models. In this talk I will begin with a short review of modern Glivenko-Cantelli theorems and inequalities for empirical processes. I will then survey some of the basic inequalities for proving consistency of MLEâ€™s, illustrated by several examples from current or recent research projects. Finally, I will review the results concerning (global) rates of convergence of MLEâ€™s due to Wong and Shen, BirgÂ´e and Massart, and van de Geer (1996). Although MLEâ€™s can be rate sub-optimal when the entropy is of sufficiently â€œhigh entropic dimensionâ€, MLEâ€™s continue to perform extremely well relative to competitors for non-parametric problems with â€œlow entropic dimensionâ€, including a number of interesting mixture models and problems with interval censoring.
The talk will conclude with several challenges and open problems.