Advanced Statistical Theory I-II

Course Description

This course focuses on drawing large sample inferences about "parameters" in statistical models. We develop asymptotic theory for maximum likelihood estimation, M-estimation, and generalized method of moment (GMM) estimation. Formal techniques for constructing estimators in semi-parametric models will be discussed. Particular attention will be paid to models for longitudinal and survival data. Guest lecturers will discuss special topics (e.g., targeted maximum likelihood, hypothesis testing, empirical likelihood). The course will involve rigorous mathematical arguments so that familiarity with concepts in advanced calculus, real analysis, and measure theory will be required.

Intended Audience

The course is designed for Biostatistics Ph.D. students in their 2nd year or beyond. Exceptions made with permission of the instructor.

Prerequisites

Introduction to Probability Theory I-II (550.620-1), Introduction to Statistical Theory I-II (140.673-4), Real Analysis.

Recommended Textbooks

Theory of Point Estimation, by E.L. Lehmann and G. Casella, Springer
A Course in Large Large Sample Theory, by T.S. Ferguson, Chapman-Hall
Elements of Large-Sample Theory, by E.L. Lehmann, Springer
Approximation Theorems of Mathematical Statistics, by R Serfling, Wiley
Asymptotic Statistics, by A.W. van der Vaart, Cambridge
Large Sample Estimation and Hypothesis Testing, by W.K. Newey and D. McFadden, Handbook of Econometrics
Theoretical Statistics, by D. Cox and D. Hinkley, Chapman and Hall
Principles of Mathematical Analysis, by W. Rudin, McGraw Hill
Counting Processes and Survival Analysis, by T. Fleming and D. Harrington, Wiley
Probability and Measure, by P. Billingsley, Wiley
Real Analysis and Probability, by R. Ash, Academic Press
Optimization by Vector Space Methods, by D. Luenberger, Wiley

Lecture Notes

Chapter 1: Brief Introduction
Chapter 2: Analysis and Asymptotics – Review
Chapter 3: Asymptotic Theory of Estimation
Chapter 4.1: Asymptotic Properties of the MLE – Part 1
Chapter 4.2: Asymptotic Properties of the MLE – Part 2
Chapter 4.3: Asymptotic Properties of the MLE – Part 3
Chapter 5: Generalized Method of Moments
Chapter 6: Hilbert Space for Random Vectors
Chapter 7: Influence Functions
Chapter 8: Semiparametric Models
Chapter 9: Stochastic Processes/Survival Analysis

Homeworks