课程大纲
COURSE SYLLABUS
1.
课程代码/名称
Course Code/Title
MAT8031
高等统计学
MAT8031 Advanced Statistics
2.
课程性质
Compulsory/Elective
专业选修课 Major Elective Courses
3.
课程学分/学时
Course Credit/Hours
3/48
4.
授课语言
Teaching Language
英文 English
5.
授课教师
Instructor(s)
蒋学军 JIANG Xuejun
6.
是否面向本科生开放
Open to undergraduates
or not
是 Open to undergraduates
7.
先修要求
Pre-requisites
（如面向本科生开放，请注明区分内容。
If the course is open to
undergraduates, please indicate the difference.）
数学分析 III(MA203a) (或数学分析精讲（MA213）)
Mathematical Analysis III (MA203a) (or Real Analysis (MA213))
数理统计 (Mathematical Statistics), 线性模型(linear models), 多元统计分析
(Multivariate statistical analysis)
8.
教学目标
Course Objectives
本课程为统计学研究生和博士生基础课，旨在帮助研究生或高年级的本科同学掌握高等统计学的基本概念和原理，为
今后从事统计学方面的研究打下扎实的基础。从概率论的原理出发，我们使用微积分，统计学概念和原理发展出统计
推断理论。本课程是理论课，将会有大量作业，强烈建议同学们独立完成，如实在有困难，可以咨询其他同学但要确
保真正理解。本课程涵盖以下主题:
1.基本概率理论; 2. 变换和期望; 3 常见的分布族; 4, 多维随机向量; 5, 随机抽样的性质; 6, 数据降维原理
; 7,
点估计和假设检验; 8, 区间估计; 9, 渐近评估.
This course serves as the fundamental course of our M.Phil. and Ph.D. programs in
statistics with the
aim of helping postgraduate students and senior
undergraduates to master some basic concepts and
theories in Advanced Statistics so as
to lay a solid foundation for the research in statistics.
Starting from the first principles of prob
ability theory, we develop the theory of statistical inference
using c
alculus, statistical concepts and principles. Home works (assignments) are essential to
understand the subject so I strongly encourage all the students try to fi nish them, please try fi
rst
independently, if there is difficulty, consult the others, but make sur
e you can do them next time
around.
This course will cover the following topics:
1. Basic probability theory; 2. Transformations and expectations; 3 Common families of
distributions;
4, Multiple random variable; 5, Properties of a random sample; 6, Princi
ple of data reduction; 7, Point
estimation and hypothesis testing; 8, Interval estimation; 9, Asymptotic evaluation.
9.
教学方法
Teaching Methods
讲授 Lectures
10.
教学内容
Course Contents
Section 1
Probability Theory Background
Section 2
Transformations and Expectations
----
moments and moment generating functions, differentiating under an
integral sign
Section 3
Common Families of Distributions
---exponential families, location and scale families, inequalities and identities
Section 4
Multiple Random Variables
----
bivariate transformation; hierarchical models and mixture distribution;
covariance and correlation; multivariate distribution
Section 5
Properties of a Random Sample
----basic concepts of random samples;
sampling from normal distribution;
order statistics; convergence concepts; generating a random variable
Section 6
Principle of Data Reduction
----
sufficient statistics; minimal sufficient statistics; complete statistics;
likelihood function; formal likelihood principle
Section 7
Point Estimation
----moments estimator; ma
ximum likelihood estimators; bayes estimators; EM
algorithm; sufficiency and unbiasedness; best unbiased estimators
Section 8
Hypothesis Testing
----likelihood ratio tests; bayesian tests; error probability;
power function;
most powerful tests; p-values; loss function optimality
Section 9
Interval Estimation
----inverting a
test statistic; pivotal quantities; pivoting the CDF; bayesian
intervals; size and converge probability; test related optimality; loss function
optimality
Section 10
Asymptotic Evaluations
----consistency and efficiency; bootstrap standard errors; M-est
imators;
Asymptotic distribution of LRTs; other large sample tests; approximate
maximum likelihood intervals; other large sample intervals
…………
11.
课程考核
Course Assessment
（○
1
考核形式
Form of examination
；○
2 .
分数构成
grading policy
；○
3
如面向本科生开放，请注明区分内
容。 If the course is open to undergraduates, please indicate the difference.）
课堂表现 Class Performance 5%
平时作业 Assignments 25%
期中考试 Mid- Term Test 30%
期末考试
Final Exam 40%
12.
教材及其它参考资料
Textbook and Supplementary Readings
[1] Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation (2nd Edition). Springer Texts in
Statistics, Springer-Verlag, New York. [Each student will be provided an e-book of this monograph]
[2] Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer Texts in Statistics, Springer-
Verlag, New York. [Each student will be provided an e-book of this monograph]
