Statistical Inference

Discussion Group

Author

Affiliation

Chi-Kuang Yeh

Georgia State University

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor.

1 Meeting time

• 9:00 - 11:00 Tuesday (Online) and 10:00 - 12:00 Friday

2 2025 Timeline

Status	Date	Notes	Topic
✅ Complete	Sep 26	1	Multivariate Normal
✅ Complete	Oct 3	2	\(\chi^2\) distribution
✅ Complete	Oct 10	3	Asymptotic modes
✅ Complete	Oct 14	4	Big O pee and small O pee
✅ Complete	Oct 17	5	Relationship of Big O and Small O, with convergence modes
✅ Complete	Oct 21	6	CMT and Slutsky’s Theorem
✅ Complete	Oct 24	7	Wald point estimation
✅ Complete	Oct 28	8	Wald confidence interval
✅ Complete	Oct 31	-	Georgia Statistics Day
✅ Complete	Nov 4	9	Asymptotic Distribution of \(\hat{\theta}_n\)
✅ Complete	Nov 7	-	Colloquium
✅ Complete	Nov 11	10	Likelihood Ration Test
✅ Complete	Nov 14	11	Regularity conditions of MLE
✅ Complete	Nov 18	12	Moments
✅ Complete	Nov 21	13 -14 (1st half)	Quantile
⏳ In Progress	Nov 24-28	Thanksgiving
🔜 Planned	Dec 2	14 (2nd half)	Quantile and Bahadur’s Representation

3 Table of Content

Chapter 0 Multivariate Normal Distribution

• 0.1 Definition

• 0.2 Properties of \(N_p(\mu,\Sigma)\)

• 0.3 Quadratic form of \(N_P(\mu,\Sigma)\)

Chapter 1 Review of Probability Theory

• 1.1 \(\sigma\)-algebra and Probability Function

• 1.2 Random Variables

• 1.3 Expectation

Chapter 2 Asymptotic Theory

• 2.1 Mode of Convergence

• 2.2 Convergence in Distribution and Central Limit Theorem

• 2.3 Stochastic Order: Big \(O\) and small \(o\)

• 2.4 Convergence of Transformation and Slutsky’s Theorem

Chapter 3 Parametric Likelihood Method

• 3.1 Strong Consistency of MLE

  • Wald point estimation
  • Wald confidence interval

• 3.2 Asymptotic Distribution of \(\hat{\theta}_n\)

• 3.3 Limiting Distribution of Likelihood Ration Test

Chapter 4 Empirical CDF, Moments and Quantile

• 4.1 Empirical cdf and properties

• 4.2 Sample Moments

• 4.3 Sample Quantiles

  • Definition and Properties
  • Point Convergence
  • Limiting Distribution
  • Bahadur’s Representation

Chapter 5 Empirical Likelihood Method

• 5.1 Ideas for Empirical Likelihood

• 5.2 Empirical likelihood for mean of population

  • Main results
  • Numerical Implementation
  • Proof of main results

• 5.3 Empirical likelihood for estimating equation

Chapter 6 Bootsrap

• 6.1 Introduction

• 6.2 Nonparametric Bootstrap method

• 6.3 Bootstrap Estimation of sampling distribution

• 6.4 Bootstrap confidence interval

• 6.5 Some theoretical results

  • Consistency property of \(H_{Boot}(x)\)
  • consistency property of \(V_{boot}\)
  • Asymptotic results for Bootstrap CI

4 Credit

The notes is based on Prof. Pengfei Li from UWaterloo’s lecture notes for STAT 908 course.