# Statistics 701 Mathematical Statistics II

### MW 5-6:15pm         Rm MTH 0304             Spring 2015

This course introduces mathematical statistics at a theoretical graduate level,                               Handouts.
using tools of advanced calculus and basic analysis. The objectives are to treat
diverse statistically interesting models for data in a conceptually unified way;                    Homework Assignments.
to define mathematical properties which good procedures of statistical inference
should have; and to prove that some common procedures have them.                                    Homework Solutions.

Prerequisite: Stat 700 or {Stat 420 and Math 410}, or equivalent.

Text:   (Required)    George Casella and Roger Berger Statistical Inference,   2nd ed., Duxbury, 2002.
(Recommended) V. Rohatgi and A.K. Saleh, An Introduction to Probability and Statistics, 2nd ed., Wiley, 2001.
Jun Shao, Mathematical Statistics, 2nd ed., Springer, 2003.
P. Bickel & K. Doksum Mathematical Statistics, vol.I, 2nd ed., Pearson Prentice Hall, 2007.
P. Billingsley, Probability and Measure, 2nd (1986) or later edition, Wiley.

A sheet of errata in Casella & Berger compiled by Jeffrey Hart of Texas A&M Stat Dept can be found here.
The most important of these errata are the ones on p.288, in Equation (6.2.7) and in line 14 (the last line
of Thm. 6.2.25): what is important as a sufficient condition for completeness is that the set of "natural parameter"
values   (η1,...,ηk) = (w1(θ),w2(θ),...,wk(θ))   fills out an open set in   Rk   as   θ   runs through all of   Θ.

Course Coverage: Last fall, Stat 700 covered the Casella & Berger through Sec.8.2.2, omitting primarily the topics involving large-sample asymptotics. This term, we begin with Sections 5.5.3 and 5.5.4, then discuss asymptotic topics related to the comparison of large-sample variances of method of moments and ML estimators (Chapter 7, Sec.2). We will re-visit the topic of multidimensional-parameter) Fisher Information along with numerical maximization of likelihoods, via Newton-Raphson, Fisher scoring, and EM algorithm. We complete this part of the course by proving that MLE's under regularity conditions are consistent and asymptotically normal, with related facts about the behavior of the Likelihood in the neighborhood of the MLE. For rigorous development of this material, we follow roughly the development of Chapter 5 of the book of Bickel and Doksum, although this material is covered non-rigorously in the univariate case in Casella and Berger's Section 10.1.
Next we return to the material not covered last term in Chapters 8 (Hypothesis Testing) and 9 (Confidence Intervals), incorporating large-sample asymptotics through the notions of asymptotic size and power, asymptotically pivotal quantity, and asymptotic confidence level. Likelihood ratio testing in composite-parameter settings will be covered in some detail, following Section 8.2.1 for the finite-sample ideas. The asymptotic aspects of all of these topics are introduced in Chapter 10 of Casella-Berger, but done more comprehensively in Bickel and Doksum's Chapter 5 and 6, the latter covering multidimensional parameters and the relation between Wald tests, score tests, and likelihood ratio tests.
In Casella and Berger, we cover through the end of Chapter 10, and in Bickel-Doksum, Chapters 5-6 including some discussion of misspecified models, `M-estimation', and estimating equations. Additional topics to be covered include: the Bootstrap and its relation to the estimation of variances and confidence intervals; Bayes and empirical Bayes methods including a brief introduction to Markov Chain Monte Carlo computational methods. All of these topics not covered in Casella-Berger or Bickel-Doksum are included in the book of Shao, but some of these topics are covered in this course primarily through handouts.
Lecture Topics by Date.

Grading: There will be graded homework sets roughly every 1.5--2 weeks (7 altogether). There will be two in-class
tests, the first on Monday March 2, covering material through the proofs about MLE asymptotics, the second tentatively
scheduled for April 29, and an in-class Final Exam. The course grade will be based 30% on homeworks, 40% on tests,
and 30% on the Exam.
No Test make-ups will be offered. Homework will generally not be accepted late, and must be handed in as hard-copy (except for
people off-campus when the due-dates are Fridays, from whom electronic-format HW will be accepted as email attachments).

HONOR CODE

The University of Maryland, College Park has a nationally recognized Code of Academic Integrity, administered by the Student Honor Council. This Code sets standards for academic integrity at Maryland for all undergraduate and graduate students. As a student you are responsible for upholding these standards for this course. It is very important for you to be aware of the consequences of cheating, fabrication, facilitation, and plagiarism. For more information on the Code of Academic Integrity or the Student Honor Council, please visit http://www.shc.umd.edu.

To further exhibit your commitment to academic integrity, remember to sign the Honor Pledge on all examinations and assignments:
"I pledge on my honor that I have not given or received any unauthorized assistance on this examination (assignment)."

Office Hours: My office hours will be Monday 4-5 and Wednesday 11-12.  I will often be available also at other times,
except on Tuesdays and Thursdays, but please send an e-mail or arrange with me in class for an office appointment.

### This course web-page will serve as the Spring 2015 Course Syllabus for Stat 700. Also: messages and updates (such as corrections to errors in stated homework problems or changes in due-dates) will generally be posted here, on this web-page, and only in rare cases will be made through emails in the course "Reflector" account, which is stat701-0101-spr15@coursemail.umd.edu . Students can also use this course email to post comments or questions that the whole class will see, and which I will answer in the same way.   Additional information:     timing of individual lectures and tests;         Important Dates below;                   for auxiliary reading, several useful handouts that are described and linked below;            cumulatively updated copy of all homework problem assignments;          partial problem set solutions posted throughout the term.

(I)   A handout on conjugate priors for a class of exponential family densities and probability mass functions.

(III)   Lecture Notes from Stat 705 on Numerical Maximization of Likelihoods. Some further steps in R showing
numerical maximization of a Gamma likelihood can be found in this class handout.

(IV)    Union-Intersection Tests covered in Casella and Berger are discussed in a journal article in
connection with applications to so-called Bioequivalence trials.

(V)   Summary of calculations in R comparing three methods for creating (one-sided)
confidence intervals for binomial proportions in moderate sized samples.

(VI).   Handout containing single page Appendix from Anderson-Gill article (Ann. Statist. 1982)
showing how uniform law of large numbers for log-likelihoods follows from a pointwise strong law.

(VII).   Handout on the 2x2 table asymptotics covered in a 2009 class concerning different sampling
designs and asymptotic distribution theory for the log odds ratio.

(VIII).   Handout on Wald, Score and LR statistics covered in class April 10 and 13, 2009.

(IX).   Handout on Chi-square multinomial goodness of fit test.

(X)   Handout on Proof of Wilks Thm and equivalence of corresponding chi-square statistic with
Wald & Rao-Score statistics which will complete the proof steps covered in class.

(XI) Background on Markov Chain Monte Carlo: First see Introduction and application of MCMC
within an EM estimation problem in random-intercept logistic regression. For additional pdf files of
"Mini-Course" Lectures, including computer-generated figures, see Lec.1 on Metropolis-Hastings Algorithm,
and Lec.2 on the Gibbs Sampler, with Figures that can be found in Mini-Course Figure Folders.

(XII).   A DIRECTORY OF SAMPLE PROBLEMS FOR old IN-CLASS FINAL (with somewhat different
coverage) CAN BE FOUND HERE. Similarly, SAMPLE FOR old IN-CLASS TESTS CAN BE FOUND HERE .
A handout of topics and sample problems given for the March 31, 2014, in-class test, can be found here.

Other handout topics from Stat 700, including Sample Problems and Tests on Stat 700 topics, Distributions
determined by moments
, Mixture and models, Topics on Statistical Simulation.

See the Resources page at the UMCP Stat Consortium.

### Important Dates

• First Class: January 26
• First Mid-Term Exam: Wed., March 11
• Spring Break: March 16-20
• Second In-Class Exam: Mon., May 4
• Last Day of Classes: Mon., May 11
• Review Session for Exam: Wed., May 13, in-class, 4-6 pm
• Final Examination: Mon., May 18, 4-6 pm