Topics in Statistical Theory

Description：

This course will recover research results of the Instructor over the years:

1) Elegant nonparametric, minimum distance estimation of a density and a regression type function and of their derivatives, with upper and lower rates of convergence; the parameter space is assumed to be either totally bounded or regular. Plug-in upper convergence rates for estimates of a mixing density in Rd and for its derivatives.

2) Rates of convergence of estimates, Kolmogorov’s entropy and the dimensionality reduction principle in regression.

3) Pathologies of the Wasserstein distance in Statistical Inference.

4) Shrinkage of U-statistics obtained with artificial augmentation of the sample size. Pitman's closeness criterion and shrinkage estimates of the variance and the SD.

5) On Tukey’s poly-efficiency.

6) Pathologies of the Bootstrap.

7) Pathologies of the MLE, with correction using Model Updated MLE (MUMLE) with DECK-principle; D=Data E=Evolves, C=Creates, K=Knowledge. Relation of MUMLE with Wallace’s Minimum Message length method.

8) Additional topics if time permits.

Prerequisite：

A course in Mathematical Statistics and Probability, including modes of convergence of random variables/vectors.