# Statistics 200C

# Theoretical Statistics

### Spring Quarter, 2010

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

Th at 1:00, 6201 Math Sci.

#### 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

### Chapman & Hall, 1996.

Table of Contents

#### 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.