Advanced Theory of Statistics
This course covers theoretical and applied fundamentals of statistical inference for beginning graduate students in statistics and probability. The main contents include statistical models, estimation, hypothesis testing, and asymptotic theory.
Lecturer
Date
19th September ~ 31st December, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 15:20 - 16:55 | A14-202 | ZOOM A | 388 528 9728 | BIMSA |
Prerequisite
Calculus, Fundamentals of probability and statistics
Syllabus
Sampling Theory
Some special statistics
Distributions of special statistics
Point Estimation
Confidence Intervals for Means
Confidence Intervals for Differences of Means
Tests of Statistical Hypotheses
Additional Comments About Statistical Tests
Chi-Square Tests
Measures of Quality of Estimators
A Sufficient Statistic for a Parameter
Properties of a Sufficient Statistic
Completeness and Uniqueness
The Exponential Class of Distribution
Functions of a Parameter
The Case of Several Parameters
Minimal Sufficiency and Ancillary Statistics
Sufficiency, Completeness and Independence
Bayesian Estimation
Fisher Information and Rao-Cramér Inequality
Limiting Distributions of Maximum Likelihood
Certain Best Tests
Uniformly Most Powerful Tests
Likelihood Ratio Tests
The Distribution of Quadratic Forms
A Test of the Equality of Several Means (One-way ANOVA)
Noncentral chi-square and Noncentral
Analysis of Variance (ANOVA)
Some special statistics
Distributions of special statistics
Point Estimation
Confidence Intervals for Means
Confidence Intervals for Differences of Means
Tests of Statistical Hypotheses
Additional Comments About Statistical Tests
Chi-Square Tests
Measures of Quality of Estimators
A Sufficient Statistic for a Parameter
Properties of a Sufficient Statistic
Completeness and Uniqueness
The Exponential Class of Distribution
Functions of a Parameter
The Case of Several Parameters
Minimal Sufficiency and Ancillary Statistics
Sufficiency, Completeness and Independence
Bayesian Estimation
Fisher Information and Rao-Cramér Inequality
Limiting Distributions of Maximum Likelihood
Certain Best Tests
Uniformly Most Powerful Tests
Likelihood Ratio Tests
The Distribution of Quadratic Forms
A Test of the Equality of Several Means (One-way ANOVA)
Noncentral chi-square and Noncentral
Analysis of Variance (ANOVA)
Reference
Introduction to Mathematical Statistics by Robert V. Hogg and Allen T. Craig
Statistical Inference by Casella and Berger
Mathematical Statistics by J. Shao
Statistical Inference by Casella and Berger
Mathematical Statistics by J. Shao
Audience
Advanced Undergraduate
, Graduate
Video Public
No
Notes Public
No
Language
Chinese
, English
Lecturer Intro
Xin Zhao is currently an Associate Professor of Mathematics at Beijing Institute of Mathematical Sciences and Applications (BIMSA). Her research interests lie at the intersection of biomathematics, computational biology, and artificial intelligence, with a focus on developing mathematical models, computational approaches, and AI-driven methods to address complex problems in the life sciences. She has authored an academic monograph published by Springer Nature and has published several SCI-indexed research articles in related fields. As the principal investigator, she has completed a National Natural Science Foundation of China (NSFC) Young Scientists Fund project and two Tsinghua University Qingfeng projects. As a key contributor, she has participated in multiple research projects funded by the NSFC, including major research programs, key projects, and general programs, as well as projects supported by the Beijing Natural Science Foundation. She have extensive teaching experience in mathematics and statistics, including courses such as Mathematical Analysis, Advanced Algebra, Probability Theory, Mathematical Statistics, and Multivariate Statistical Analysis. She is currently a core member of the Beijing Strategic Scientists Team. Her academic honors include the Outstanding Doctoral Graduate of Tsinghua University, Outstanding Doctoral Dissertation Award of Tsinghua University, National Scholarship, First-Class Comprehensive Excellence Scholarship and Ruo lin award.