STAT 5010

Advanced Statistical Inference

Class Information

- Class time: M 1830-2115
- Location: YIA LT7
- Outline: 2024Fall_S5010_outline.pdf
- Password: see Blackboard

Instructor

- Name: Kin Wai CHAN
- Email: kinwaichan@cuhk.edu.hk
- Office: LSB 115
- Tel: 3943 7923
- Office hour:
  I have an open-door policy. Feel free to drop by anytime and ask me questions.

Teaching Assistants

Zhao, Yikun

- Email: Zyk3355601STAT@link.cuhk.edu.hk
- Office: LSB 145
- Tel: 3943 3230

Keung, Wing Tung (Toto)

- Email: wtkeung@link.cuhk.edu.hk
- Office: LSB 123
- Tel: 3943 8522

Lee, Chak Ming (Martin)

- Email: 1155125610@link.cuhk.edu.hk
- Office: LSB 130
- Tel: 3943 7939

Description

This course is concerned with the fundamental theory of statistical inference. It covers statistical models, point estimation, set estimation, hypothesis testing, decision theory, large sample theory, methods for evaluating inference procedures, and computational strategies.

Note: No prerequisite but knowledge of Stat 2001, 2005, 2006, 4003, and 4010 is strongly recommended.

Textbooks

A self-contained lecture note is the main source of reference. Complementary textbooks include

- - (Major) Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury Press.
  - (Minor) van der Vaart, A. W. (2000). Asymptotic Statistics. Cambridge.
  - (Minor) Lehmann, E. L. and Casella, G. (1998), Theory of Point Estimation. Springer.
  - (Minor) Lehmann, E. L. and Romano, J. P. (2005), Testing Statistical Hypotheses. Springer.
  - (Minor) Wasserman, L. (2013). All of statistics: a concise course in statistical inference. Springer.

Learning outcomes

Upon finishing the course, students are expected to

understand the foundation of statistical models and statistical inference;
provide intuitive interpretations and statistical insights on various statistical inference problems;
derive and compute statistical inference procedures based on different principles and methods; and
evaluate and compare different statistical inference procedures.

Assessment and Grading

There are three main assessment components, plus a bonus component.

- - a (out of 100) is the average score of approximately eight assignments with the lowest two scores dropped;
  - m (out of 100) is the score of mid-term project/test; and
  - f (out of 100) is the score of final project/exam.
  - b (out of 2) is the bonus points, which will be given to students who actively participate in class.

The total score t (out of 100) is given by

t = min{100, 0.3a + 0.2max(m,f) + 0.5f + b}

If min(t, f ) < 30, the final letter grade will be handled on a case-by-case basis. Otherwise, your letter grade will be in the A range if t ≥ 85, at least in the B range if t ≥ 65, at least in the C range if t ≥ 55.

* For the most updated information, please always refers to the course outline announced by the course instructor in Blackboard, which shall prevail the above information if there is any discrepancy.

Lecture Notes

* Click (S5010/2024Fall/lecture) to download lecture notes.
* The finalized version of the lecture notes will be uploaded one day before the lecture.
* All rights reserved by the authors. Re-distribution by any means is strictly prohibited.

Front Matters

- Contents
- Instructions

Part I: Basic Probability and Statistics

- § 1 Basic Probability: (a) random variables, (b) quantile function, (c) probability inequalities, (d) representation
- § 2 Limit Theorems in Statistics: (a) stochastic convergence, (b) law of large number, (c) central limit theorem, (d) Delta method

Part II: Data Reduction

- § 3 Statistical Models: (a) exponential family, (b) location-scale family, (c) identifiable family.
- § 4 Sufficiency Principle: (a) minimal sufficiency, (b) factorization theorem, (c) ancillary, (e) completeness.
- § 5 Likelihood Principle: (a) likelihood function, (b) discussion

Part III: Point Estimation

- § 6 Qualities of Point Estimators: (a) mean squared error, (b) consistency, (c) efficiency, (d) Fisher information, (e) Cramer-Rao lower bound
- § 7 Methods of Point Estimation: (a) method of moment, (b) unbiasedness, (c) maximum likelihood

Part IV: Hypothesis Testing

- § 8 Qualities of Hypothesis Tests: (a) size and level, (b) power (c) p-value
- § 9 Methods of Hypothesis Tests: (a) most powerful test, (b) likelihood ratio test, (c) Wald test, (d) Rao score test

Part V: Interval Estimation

- § 10 Qualities of Interval Estimators: (a) coverage probability, (b) expected width
- § 11 Methods of Interval Estimation: (a) Pivotal quantity, (b) inversion of tests, (c) bootstrapping

Appendixes

- § A: Basic Mathematics: (a) mathematics notations, (b) differentiation, (c) integration, (d) Taylor’s Expansion
- § B: Solution to end-of-chapter examples (It may contain errors. Please read with care.)
- P.S.: Not all materials in the appendices are directly useful for this course. I will tell you which parts are useful when we need them.

Assignments

* Click (S5010/2024Fall/A) to download assignments.

- Assignment 1: Mode of convergence and limit theory in statistics --- Due: 27 Sep (Fri) @1800
- Assignment 2: Exponential family (ZIP and Gamma-Poisson models) and identifiable family --- Due: 4 Oct (Fri) @1800
- Assignment 3: Sufficiency principle, SS, MSS, CSS, AS (multinomial model, linear regression model) --- Due: 14 Oct (Mon) @ 1800
- Assignment 4: UMVUE for Pareto data --- Due: 15 Nov (Fri) @ 1800
- Assignment 5: MLE for Pareto data --- Due: 18 Nov (Mon) @ 1800
- Assignment 6: UMPT and asymptotic tests--- Due: 27 Nov (Wed) @ 1800
- Assignment 7: Region estimation --- Due: 29 Nov (Fri) @ 1800

In-class notes

* Click (S5010/2024Fall/inclassNote) and (S5010/2024Fall/recording) to download in-class notes and recordings (if any).
* In-class notes and recordings (if any) will be uploaded within one week after the lecture

- Lecture 1 (2 Sep) --- Basic probability and modes of convergence
- Lecture 2 (9 Sep) --- Relationship between modes of convergence, limit theory in statistics
- Lecture 3 (16 Sep) --- Examples and Proofs of limit theory, Exponential family, properties of EF
- Lecture 4 (23 Sep) --- Identifiability of missing data model, Statistics, SS, MSS, AS, CS
- Lecture 5 (30 Sep) --- More about CS, proofs of theorems in Chapter 3, likelihood principle
- Lecture 6 (7 Oct) --- Quality of point estimators, MSE, examples, consistency
- Lecture 7 (14 Oct) --- CRLB, MME, QME, UMVUE
- Lecture 8 (21 Oct) --- Midterm, high-order moments, CLT for sample quantiles
- Lecture 9 (28 Oct) --- UMVUE, proof of Rao-Blackwell theorem and Lehmann-Scheffe theorem, MLE, consistency of M-estimator, CLT for MLE
- Lecture 10 (4 Nov) --- CLT for MLE, Neyman-Scott problem, Hodges' estimator, hypothesis, test, comparison, power, size, level
- Lecture 11 (11 Nov) --- p-value, power calculation, UMPT, examples
- Lecture 12 (18 Nov) --- Proof of NP lemma and KR theorem, exact LRT, asymptotic LRT, WT and RST
- Lecture 13 (25 Nov) --- Proof of asymptotic limit of LRT statistic, confidence region, pivotal quantity, duality, inversion of test
  - - Due to technical reasons, the last part of the lecture was not recorded successfully.

Quizzes

* Click (S5010/2024Fall/quiz) to download quizzes.

- Quiz 1: Exponential family, location-and-scale family, identifiable family

Mid-term exam

Start time: 21 October (Monday) @ 6:35 pm
Duration: 90 minutes
Scope: Chapter 1 -- Section 7.3 of Chaper7 (everything until and including QME)
Instructions: Please refer to the first page of the mock paper
Mock exam: S5010/2024Fall/M0
Exam and solution: S5010/2024Fall/M

Final exam

Start time: 2 December (Monday) @ 6:35 pm
Duration: 150 minutes
Scope: Chapters 1--11 (including lecture notes and in-class hand-written notes)
Instructions: Please refer to the first page of the mock paper
Mock exam: S5010/2024Fall/F0
Seating plan: S5010/2024Fall/seat
Exam: S5010/2024Fall/F

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