MATH 6740: Introduction to Mathematical Statistics (Spring 2012)

Instructor: Marten Wegkamp

The goal of this course is to acquire understanding of basic (large sample) theory in mathematical statistics.

I will use my own lecture notes. Some of it will be based on Asymptotic statistics by A.W. van der Vaart.

Selected topics in Mathematical Statistics:

Consistency of M-estimators
Wald's method of consistency
Asymptotic normality via quadratic approximation

Distances between probability measures.
-Total variation distance
-Hellinger distance

Hellinger differentiability
-Generalized information bound

Maximal inequalities
-Rates of convergence of Maximum Likelihood estimators
-Likelihood Ratio tests

Contiguity
-Limit distributions under contiguous alternatives
-perfect tests
-LeCam's lemmas

Efficiency
-Super-efficiency
-Local Asymptotic Normality
-Bahadur efficiency
-convolution theorem

Nonparametric density estimation
-Universal consistency of the histogram estimator
-Universal consistency of the kernel estimator
-mean squared errors
-cross validation: selection of the bandwidth

Bootstrap
-bootstrap principle
-jackknife
-pivotal method
-bootstrap confidence intervals

U-statistics
-Hoeffding decomposition

Rank statistics
-Asymptotic normality
-Rank tests