# Statistics 700 Mathematical Statistics I

STUDENTS: Please submit your online evaluation of the course, the text, and the instructor by going to the URL for the
CoursEvalUM page at https://www.CourseEvalUM.umd.edu or by directly logging in at https://umd.bluera.com/UMD/.

### MW 5:15-6:30pm         Rm MTH 0307             Fall 2014

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 410 or equivalent. You should be comfortable (after review) with joint densities,
(multivariate, Jacobian) changes of variable, moment generating functions, and conditional expectation.

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: This fall, we will cover the material in Chapters 1--4 fairly rapidly, as review (in about four lectures). After that, we will cover most of Chapters 5-8. We spend some time on Chap. 5 (going lightly over Sections 5.5.1-5.5.3, all other sections thoroughly), covering Chapters 6-7 in full detail and going as far through Chapter 8 as we have time for. This means that we will discuss properties of statistical procedures that are meaningful for small or moderate samples reserving asymptotic techniques for large-sample data until the spring term, and will revisit the Likelihood Ratio Test (going beyond the treatment in Casella and Berger) for large-sample data. Further coverage of Bayes and empirical Bayes methods will be presented in the spring along with material on confidence intervals. Computational topics will be treated both terms, with EM algorithm and other topics related to likelihood maximization and basic simulation in the fall, and bootstrap and Markov Chain Monte Carlo in the spring. Many of these topics will be covered through handouts, during both terms.
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 Wednesday October 8, covering material through Chapter 5, 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 11-12 and Wednesday 4-5:15.  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.

(II)    I have a paper on the topic of distributions related to the normal that are or are not uniquely determined by their moments.
The paper uses many of the techniques we review in Chapters 2 to 4.

(III)   The topic of mixture distributions and densities and their relation to hierarchical specification
of a distributional model and to distribution functions of mixed type is elaborated in this handout.

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

(VI)   Lecture Notes from Stat 705 on Numerical Maxmimization of Likelihoods.

(VII) Topics on Statistical Simulation: There are two sorts of handouts on Simulation methods and
interpretation. First, under this heading, there are 4 pdf writeups on Random Number Generation,
simulation, and interpretation of simulation experiments: (i) Pseudo-random number generation,
(ii) Transformation of Random Variables, (iii) Statistical Simulation , and if you want to read a little more
on computational speedups in statistical simulations, click here. Topics (i) and (iv) were taken from my
web-pages for the course STAT 705 on Statistical Computing in R. Additional material on
statistical simulation for Bayesian MCMC is discussed under heading (VI) below.

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

(IX) 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.

### Important Dates

• First Class: September 3
• First Mid-Term Exam: Wed., October 8
• Withdrawal Date: November 11
• Thanksgiving: November 27      (Class will be held as usual the day before.)
• Second In-Class Exam: Wed., December 3
• Review Session for Exam: Fri., December 12, in-class, 4-6 pm
• Final Examination: Thurs., December 18, 4-6 pm