**Course: MATH 541B, Graduate Mathematical Statistics II, Fall 2023**

**Prerequisite:** Math 541A

**Course Content:** Hypotheses testing, Neyman-Pearson lemma, generalized likelihood ratio procedures, confidence intervals, consistency, power, jackknife and bootstrap. Monte Carlo Markov chain methods, hidden Markov models.

*Last update:* 9 May 2023

**Instructor:** Steven Heilman, stevenmheilman(@-symbol)gmail.com

**Office Hours:** Tuesdays, 9AM-11AM, on zoom [link posted in blackboard]

**Lecture Meeting Time/Location:** Mondays, Wednesdays and Fridays 11AM-1150AM, KAP 140

**TA:** Xinze Du, xinzedu(@-symbol)usc.edu

**TA Office Hours:** (see the schedule in the Math Center)

**Recommended Textbook:** Cassella and Berger, __Statistical Inference__, 2nd Edition.

**Other Textbooks (not required):** Lehmann and Romano, __Testing Statistical Hypotheses__.

Ferguson, __A Course in Large Sample Theory__.

Shao and Tu, __The jackknife and bootstrap__.

McLachlan and Krishnan, __The EM Algorithm and Extensions__.

Robert and Casella, __Monte Carlo and Statistical Applications__.

Haggstrom, __Finite Markov Chains and Algorithmic Applications__.

Levin and Peres, __Markov Chains and Mixing Times__.

**Midterm 1:** Friday, September 29, 11AM-1150AM KAP 140

**Midterm 2:** Monday, October 30 , 11AM-1150AM KAP 140

**Final Exam:** Wednesday, December 6, 11AM-1PM, KAP 140

**Other Resources:** An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments

**Email Policy:**

- My email address for this course is stevenmheilman(@-symbol)gmail.com
- It is your responsibility to make sure you are receiving emails from stevenmheilman(@-symbol)gmail.com , and they are not being sent to your spam folder.
- Do NOT email me with questions that can be answered from the syllabus.

**Exam Procedures:** Students must bring their USCID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the USC Student Conduct Code. (See also here.)

**Accessibility Services:** If you are registered with accessibility services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Accessibility Services (OSAS) each semester. A letter of verification for approved accommodations can be obtained from OSAS. Please be sure the letter is delivered to me as early in the semester as possible. OSAS is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.

https://osas.usc.edu/

213-740-0776 (phone)

213-740-6948 (TDD only)

213-740-8216 (fax)

OSASFrontDesk@usc.edu

Discrimination, sexual assault, and harassment are not tolerated by the university. You are encouraged to report any incidents to the *Office of Equity and Diversity* http://equity.usc.edu/ or to the *Department of Public Safety* http://capsnet.usc.edu/department/department-public-safety/online-forms/contact-us. This is important for the safety whole USC community. Another member of the university community - such as a friend, classmate, advisor, or faculty member - can help initiate the report, or can initiate the report on behalf of another person. *The Center for Women and Men* http://www.usc.edu/student-affairs/cwm/ provides 24/7 confidential support, and the sexual assault resource center webpage sarc@usc.edu describes reporting options and other resources.

**Exam Resources:** Here is a page with some practice exams (at the bottom). Here is a page with some practice exams (see Assignment 3). Here is a page containing USC Stats B Qual Exams with solutions. Here is another page with some related Stats qualifying exams (see e.g. the Theory exams).

**Homework Policy:**

- Homeworks are due at
**9AM on Fridays**. - Homeworks are submitted in blackboard, under the "Assignments" tab. You are allowed unlimited submission "attempts" for an assignment, but only the last submission will be graded. To avoid internet issues, I recommend making your first submission of an assignment well in advance of the deadline.
- Homeworks should be submitted as single PDF documents. One way to create a PDF document from paper homework assignments is the freely available Adode Scan App
- Late homework is not accepted.
- If you still want to turn in late homework, then the number of minutes late, divided by ten, will be deducted from the score. (The time estimate is not guaranteed to be accurate.)
- The lowest homework score will be dropped. This policy is meant to account for illnesses, emergencies, etc.
- Do not submit homework via email.
- A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
- You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help.
- All homework assignments must be
**written by you**, i.e. you cannot copy someone else's solution verbatim. However, I would very much encourage you to form study groups and do the homework together in small groups. Homework is the most important part of a graduate mathematics course, and I encourage you to take it very seriously. - Homework solutions will be posted on Friday after the homework is turned in.

**Grading Policy:**

- The final course grade is weighted as the larger of the following two schemes. Scheme 1: homework (25%), the first midterm (20%), the second midterm (25%), and the final (30%). Scheme 2: homework (25%), largest midterm grade (35%), final (40%). The grade for the semester will be curved. However, anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
- If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
- You must attend the final exam to pass the course.

**Tentative Schedule**: (This schedule may change slightly during the course.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

1 | Aug 21: 8.1, Hypothesis Testing | Aug 22 | Aug 23: 8.1, Hypothesis Testing | Aug 24 | Aug 25: 8.2, Likelihood Ratio Test |

2 | Aug 28: 8.3.1, Neyman-Pearson Lemma | Aug 29 | Aug 30: 8.3.2, Karlin-Rubin Theorem | Aug 31 | Sep 1: Homework 1 due. 8.3, Exponential Families |

3 | Sep 4: No class | Sep 5 | Sep 6: 8.3, Unbiasedness | Sep 7 | Sep 8: 8.3.4, p-values |

4 | Sep 11: 8.3.5, Loss Function Optimality | Sep 12 | Sep 13: 9.1, Confidence Intervals | Sep 14 | Sep 15: Homework 2 due, 9.2, Test Inversion |

5 | Sep 18: 9.2.2, Pivotal Method | Sep 19 | Sep 20: 9.2.3, Pivoting CDF | Sep 21 | Sep 22: 9.2.4, Bayesian Intervals |

6 | Sep 25: 9.3.4, Loss Function and CI | Sep 26 | Sep 27: 10.3, Asymptotics of Likelihood Ratio | Sep 28 | Sep 29: Exam 1 |

7 | Oct 2: 10.3, Asymptotics of GLR | Oct 3 | Oct 4: 10.3, Asymptotics of GLR | Oct 5 | Oct 6: Homework 3 due, Jackknife |

8 | Oct 9: Jackknife | Oct 10 | Oct 11: Bootstrap | Oct 12 | Oct 13: No class |

9 | Oct 16: Bootstrap | Oct 17 | Oct 18: Consistency of Jackknife | Oct 19 | Oct 20: Homework 4 due, Bootstrap/Jackknife Relationship |

10 | Oct 23: 12.2, Analysis of Variance | Oct 24 | Oct 25: 12.2, Analysis of Variance | Oct 26 | Oct 27: 14.2, Linear Regression |

11 | Oct 30: Exam 2 | Oct 31 | Nov 1: EM Algorithm | Nov 2 | Nov 3: Homework 5 due, EM Algorithm |

12 | Nov 6: Monte Carlo | Nov 7 | Nov 8: Rejection Sampling | Nov 9: | Nov 10: No class |

13 | Nov 13: Importance Sampling | Nov 14 | Nov 15: Markov Chains | Nov 16 | Nov 17: Homework 6 due, Markov Chains |

14 | Nov 20: Metropolis-Hastings | Nov 21 | Nov 22: No class | Nov 23 | Nov 24: No class |

15 | Nov 27: Markov Chain Monte Carlo | Nov 28 | Nov 29: Review of course | Nov 30 | Dec 1: Homework 7 due, Review of course |

**Advice on succeeding in a math class:**

- Review the relevant course material
**before**you come to lecture. Consider reviewing course material a week or two before the semester starts. - When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
- Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least
**two days**before it is due, to alleviate deadline stress. - Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, I would very much recommend typesetting your homework. Learning LaTeX is a very important skill to have for doing mathematics. Here is a template .tex file if you want to get started typesetting.
- If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

**Homework**

**Homework .tex files**

**Exam Solutions**

**Supplementary Notes**