# Advanced Statistical Inference I & II

### Last updated 2/13/2006 (Established in July 2005)

Note: Prediction is difficult, especially with respect to the future. As a result, listings of plans for classes, and of homework, are always subject to change. To be sure you're preparing the right homework, doing the right readings, etc.... check this page regularly. Often, homework assignments in particular will change --- either because of a small change to the assignment, or because I didn't get as far in class as I had planned and hence have to change (usually reduce) the assignment.

Note: Class meets Wednesday 1:00 - 2:50 and Friday 11:20-12:10 in 新數102 except where otherwise noted.

Instructor: 陳宏  舊數106   Email: hchen@math.ntu.edu.tw  Phone#: 3366-2846

Office hours:  Wednesday 11-12/Friday 10:20-11:00, or by appointment

Teaching Assistant:  黃信雄           Email:  r93221018@ntu.edu.tw     Office hours:  Friday afternoon

## 成績評量方式

• 習題（20%每一章指定一次每次應該不超過10題）
• 期中考（30%
• 期末考（30%
• 小  考 (20%)

The objective of this course is to introduce to the students some basic theory of probability. It is fundamentally important for understanding  the commonly used statistical concepts and methods. It also provides a necessary basis for students for a further study of other advanced statistical courses.

Outline

Fall semester:

1. Probabilities, random variables, and distributions.
2. Transformations and expectations.
3. Common families of distributions.
4. Multivariate probability distributions and related properties.
5. Random samples, sampling distributions, and convergence concepts.
6. Sufficiency, likelihood, and equivalence principles.

Spring semester

• ### Neyman-Pearson Lema, Bayesian test, UMP test and UMPU test

Chapter 9: (2 weeks) Interval estimation

• Inverting method, pivotal quantity, p-value,

• Bayesian interval estimates

• Bayes procedure, decision and prediction theory: Loss and risk function, 23
Bayes rules, minimax principle, optimal decision rules, predictive distribution

Chapter 10: (3 weeks) Asymptotic methods

• Consistency and efficiency of MLE, Bootstrap approach

• Robustness, Huber estimator, Robustness of sample mean,
sample median, and M-estimator

• Wald and Score statistic

Chapter 11, 12: (1 week) Topics of Linear model, generalized linear model and logistic model

Text: Statistical Inference, Second Edition, Casella and Berger
• Chapters 1-6 of Casella and Berger will form the basis of the course.

1. Rice, John A. (1995). Mathematical Statistics and Data Analysis. 2nd edition. Duxbury Press.
2. Bickel, P.J., and Doksum, K.A. (2001). Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I. 2nd edition Prentice Hall.
3. Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation. 2nd Edition, Springer.

## Homework Assignment

課程內容及進度

September  October    November    December   January  February  March  April   May June

February 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 學期 上課開始 21 22 Class Sufficient Principle ch6 23 24 Class Likelihood Principle 25 26 27 28和平紀念日

• March 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 Class Equivalence Principle 2 3 Class Point Estimation 4 5 6 加退選 7 8  Class Point Estimation 9 10 Class  Point Estimation 11 12 13 14 15 Class Point Estimation 16 17 Class Point Estimation 18 19 20 21 22 Class Point Estimation 23 24Class Point Estimation  2 & 3 25 26 27 28 29  Class Asymptotic methods   4 30 31 Class Asymptotic methods 6

April 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3 4 5民族掃墓節 6 7 Class  Asymptotic methods 8 9 10 11 12  Class  Asymptotic methods 13 14 Class  Asymptotic methods 15 16 17 期中考週 18 19 Class Asymptotic methods Overview 20 21 Class Asymptotic methods 3 22 23/30 24 25 26  Class  Test of hypothesis  4 27 28 Class Test of hypothesis  6 29

May 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3  Class Test of hypothesis 4 5  Class  Test of hypothesis 6 7 8 9 10   Class Test of hypothesis 11 12 Class  Test of hypothesis 13 14 15 16 17 Class  Interval estimation 18 19  Class  Interval estimation 20 21 22 23 24 Class   linear model 25 26 Class  linear model B3 27 28 29 30 31  端午節

June 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2  Class 3 4 5 6 7   Class 8 9  Class 10 11 12 13 14   Class 15 16  Class 17 18 19 20 21 期末考 22 23 24 25 26 27 28 29 30

September 2005

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18中秋節 19 學期 上課開始 20 21 Class Overview   Set Theory 1 22 23 Class Basics of Probability Theory 2 & 3 24 25 26 27 28  Class Conditional Probability and Independence 4 29 30 Class Random Variable 5 Density and Mass Functions 6

October 2005

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 2 3 4 5 Class Distributions of Functions of a Random Variable 7 　    Summary 6 7 Class Expected values 8 8 9 10 國慶   紀念日 11 12 Class  Summary Moments and moment generating functions 9 13 14 Class Differentiating under an integral sign 15 16 17 18 19 Class  Summary Discrete Distribution 11  & 12 & 13 20 21 Class Discrete Distribution 16 22 23 24 25 26 Class  Summary Continuous Distribution 17 & 18 27 28 Class Continuous Distribution 19 29 30 30 31

### November 2005

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 Class  Exponential Families Location and Scale Families 20 3 4 Class Inequalities and Identities 21 5 6 7  期中考    試開始 8 9 Class: Review 2nd homework. Multiple Random Variables 22 10 11  Class Quiz 1 12 13 14 15 停課   一天 16 Class  Multiple Random Variables 23 17 18  Class Transformations 24 19 20 21 22 23 Midterm 24 25  Class  Hierarchical Models and Mixture Distributions 26 & 27 26 27 28 29 30 Class  Quiz 2

### December 2005

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2  Class  Multivariate Distribution 31 Convergence Concepts 40 3 4 5 6 7 Class  Summary Sums of Random Variables from a Random Sample 32 & 33 8 9  Class  Properties of the sample mean and variance 34 & 35 10 11 12 13 14 Class Order Statistics and Convergence Concepts 37 & 38 15 16 Class Generating a random sample 17 18 19 20 21 Class 22 23 Class 24 25 26 27 28 Class 29 30 Class 31

### January 2006

 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 1 2 3 4 Class  Summary 5 6 Class  Review 上課最後一天 7 8 9 10 11 期末考 12 13 14 15 16 寒假開始 17 18 19 20 21 22 23 24 25 26 27 28 除夕 29 30 31

### Some useful website:

1.  Optimization (convex and constrained optimization) See Lecture Notes 1, 2, 3, 10

at http://www.princeton.edu/~chiangm/class.html.

2.  Linear algebra: Professor Strang's Class 18.06 Linear Algebra Lecture Videos, Fall 1999