Course No.：S070100XJ016
Course Category：Basic Course of Subject
Period/Credits：40/2
Prerequisites：Mathematics Analysis, Linear Algebra, Probability Theory and Mathematics Statistics.
Aims & Requirements：
This course is the basic disciplinary course for the masters and doctors of the major of probability theory and mathematics statistics, and also can be used as a select course for the masters of the major of mathematics and other science. Statistics has rich contents, in this course we mainly teach the basic conceptions, methods, theories, computations of mathematics statistics to lay a solid foundation for the further study and work in the field of statistics. The main contents of this course include point estimators, hypothesis tests, confidence intervals(regions).
Primary Coverage：
Chapter 1 Fundamentals of Statistics
Sample space and sample distribution families, Distribution of order statistics, review of measure theory, Slutsky theorem, Exponential families, Fisher information, Statistics and its sufficiency, Neyman Factorization Criterion, Minimal sufficient statistics, Complete statistics, Basu theorem, brief introduction of Bayes method, Wald’s statistical decision theory, Admissible of estimations.
Chapter 2 Point estimation
The basic conceptions of point estimators, Unbiased estimation, Uniformly minimum variance unbiased estimation, Rao-Blackwell theorem, Lehmann-Scheffe theorem, Cramer-Rao low bound, Bayes estimators, Stein’s Phenomenon and Shrinkage Estimators, Invariant Estimators, Pitman Estimators, The Asymptotic Property of Point Estimators, Delta Method, Asymptotic relative efficiency, Moment Estimator, Maximum likelihood estimators(MLEs), Kullback-Leibler distance, The Asymptotic theory of MLEs with one dimension, Standard deviation of MLEs, Expected and observed Fisher information, Multidimensional situation, Numerical Calculation of MLEs, Newton-Raphson Algorithm, Fisher scoring Algorithm.
Chapter 3 Hypothesis Testing
Basic conceptions, Test function, Level of significance and power function, Type Ⅰerror and type Ⅱ error, Uniformly most powerful test(UMPT), Neyman-Pearson Lemma, UMPT for Monotone Likelihood ratio families, Uniformly most powerful unbiased test(UMPUT), Similar and Neyman structure, UMPUT for One Parameter Exponential Families, UMPUT for Multi-parameter Exponential Families, UMPUT for Normal Families, Invariant Test, p-value, Likelihood Ratio Test, Wilks Phenomenon, Wald Test, Score Test, Nonparametric Tests, Sign Tests, Permutation Tests, Wilcoxon Signed-rank Test, Kolmogorov-Smirnov Test, Cramer-von Mises Test, Chi-square Tests.
Chapter 4 Confidence Intervals(regions)
Basic conceptions, Criteria for comparing confidence intervals, Asymptotic Confidence Intervals(regions), Pivotal Method, Obtain confidence intervals(regions) via Hypothesis Test, Fiducial Method.
References：
[1] Mao Shisong et.al，Advanced Mathematical Statistics，Higher Education Press，1998.
[2] Chen Xiru，Advanced Mathematical Statistics，Press of University of Science and Technology of China，1999.
[3] Chen Xiru，Introduction to Mathematical Statistics，Science Press，1997.
[4] E.L. Lehmann, G.. Casella Theory of Point Estimation, 2nd edition,1998.
[5] E.L. Lehmann, Testing Statistical Hypotheses, 2nd edition,1986.
Author：Sanguo Zhang (School of Mathematical Sciences, GUCAS)
Date：June, 2009
