Humboldt-Universität zu Berlin - Statistik

The course covers a part of mathematical statistics which deals with the limiting behavior of different sample statistics, U-statistics, M-, L- and R-Estimates. This course gives better understanding for the basic tools learned in the elementary Statistics I and II, like Law of Large Numbers, Central Limit Theorem, Kolmogorov-Smirnov and Cramer-von-Mises tests, sample mean and sample variance behavior, etc. This course is laying a bridge between the probability theory and the mathematical statistics by manipulating with “probability” theorems to obtain “statistical” theorems.

In the first part of the course we discuss basic tools of asymptotic theory in statistics: convergence in distribution, in probability, almost surely, in mean. We also consider main probability limit laws: LLN and CLT. Then we deal with the usual statistics computed from a sample: the sample distribution function, the sample moments, the sample quantiles, the order statistics. Properties, such as asymptotic normality and almost sure convergence will be derived in the lecture. Afterwards, comes the asymptotics of statistics concocted as transformations of vector of more basic statistics. Next part concerns statistics arising in classical parametric inference and contingency table analysis. These include maximum-likelihood estimates, likelihood-ratio tests, etc. Last part of the course treats U-statistics, statistics obtained as solutions of equations (M-estimates), linear function of order statistics (L-estimates) and rank statistics (R-estimates).

Outline

1. Concepts of convergence and basic limit theorems
2. Sample statistics
3. Transformation of given statistics
4. Asymptotic theory in parametric inference
5. U-Stastistics
6. .M-, L- and R-Estimates

Literature

• R.J.Serfling, Approximation theorems of mathematical statistics, 1980, Wiley series in mathematics.