# Theoretical Statistics

### Spring Quarter, 2010

Time & Place: MWF at 1:00, 5203 Math Sci.
Th at 1:00, 6201 Math Sci.

## Instructor: Thomas S. Ferguson

#### Generalized Office Hour: Thurs. 1:00 pm. in 6201 Math. Sci.

Other Office Hours: Door usually open after 3:00 pm. 8917 Math. Sci. and by appointment.
E-mail: tom@math.ucla.edu

#### Generalized Reader: Jaishen Yu, office hour: 3:30-4:30 Wed.

Prerequisites: Statistics 200B or consent of instructor.
Topics: Large sample properties of tests and estimates, consistency and efficiency, U-statistics, chi-squared tests.
There will be one midterm in the sixth week. The final examination is on Thursday, June 10, from 11:30 AM to 2:30 PM.

Homework problems from Additional Exercises.: (due on Fridays)

• Exercise Set 1. Problems 1.4 and 2.1 a,b,c. Solutions.
• Exercise Set 2. Problems 2.7, 3.5 and 4.1. Solutions.
• Exercise Set 3. Problems 5.5, 5.6 and 6.3. Solutions.
• Exercise Set 4. Problems 7.8, 8.2 and 9.6. Solutions.
• Exercise Set 5. Problems 10.3, 11.3 and 12.2. Solutions.
• Exercise Set 6. Problems 12.4 and 13.2. Solutions.
• Midterm Examination and Solutions.
• Exercise Set 7. Problem 14.1. Solutions.
• Exercise Set 8. Problems 17.4, 18.6 and 19.3. Solutions.
• Exercise Set 9. Problems 20.5, 22.1 and 22.5. Solutions.
• Exercise Set 10. Problems 24.1, 24.4 and 24.6. Solutions.
• Last Year's Final Examination and Solutions.
• This Year's Final Examination and Solutions.
• Stein's Normal Approximation Theorem.
• U-Statistics.
• Distribution Function Calculators.

## Text: A Course in Large Sample Theory

#### Part 1: Basic Probability Theory.

1. Modes of Convergence.
2. Partial Converses.
3. Convergence in Law.
4. Laws of Large Numbers.
5. Central Limit Theorems.

#### Part 2: Basic Statistical Large Sample Theory

6. Slutsky Theorems.
7. Functions of the Sample Moments.
8. The Sample Correlation Coefficient.
9. Pearson's Chi-Square.
10. Asymptotic Power of the Pearson Chi-Square Test.

#### Part 3: Special Topics.

11. Stationary m-dependent Sequences.
12. Some Rank Statistics.
13. Asymptotic Distribution of Sample Quantiles.
14. Asymptotic Theory of Extreme Order Statistics.
15. Asymptotic Joint Distributions of Extrema.

#### Part 4: Efficient Estimation and Testing.

16. A Uniform Strong Law of Large Numbers.
17. Strong Consistency of the Maximum Likelihood Estimates.
18. Asymptotic Normality of the MLE.
19. The Cramer-Rao Lower Bound.
20. Asymptotic Efficiency.
21. Asymptotic Normality of Posterior Distributions.
22. Asymptotic Distribution of the Likelihood Ratio Test Statistic.
23. Minimum Chi-Square Estimates.
24. General Chi-Square Tests.