Fall 2024

**Instructor:** Professor Eric Slud,
Statistics Program, Math Dept.,
Rm 2314, x5-5469, slud@umd.edu

**Office hours: **M 10, F 11 (initially), or email me to make an appointment (can be on Zoom).

** TA/Grader in the course is Amandeep Chanda**, who will hold regular office hours Mondays 2:30-4 for students to ask questions about lectures and HW problems.

**Overview:** This course introduces mathematical statistics at a theoretical graduate level, using tools of advanced calculus and basic analysis. The framework is to define families of probability models for observed-data structures and explain the sense in which functions of observed-data random
variables can give a good idea of which of those probability models governed a particular dataset.
The course objective is to treat diverse statistically interesting models for data in a conceptually
unified way; to define mathematical properties that good procedures of statistical inference should
have; and to prove that some common procedures have them. Aspects of the theoretical results are
illustrated using demonstrations with statistical simulation. **The fall term (Stat 700) will primarily emphasize definitions, concepts and finite-sample optimality properties of estimators and tests, while the spring term (Stat 701) emphasizes large-sample asymptotic theory and more modern topics.**

**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, Central Limit Theorem and Law of Large Numbers, and mathematical analysis proofs at the
level of Math 410-411. **The first HW set, due 9/4/24, will have several problems at the level of the prerequisite review material.**

**Required Course Text:** P. Bickel & K. Doksum, *Mathematical Statistics, vol.I,
2nd ed.*, Pearson Prentice Hall, 2007.

**Recommended Texts:** *(i)* George Casella and Roger Berger
*Statistical Inference*, 2nd ed., Duxbury, 2002.

*(ii)* V. Rohatgi and A.K. Saleh, *An Introduction to Probability and Statistics*, 2nd ed., Wiley, 2001.

*(iii)* Jun Shao, * Mathematical Statistics*, 2nd ed., Springer, 2003.

*(iv)* 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 (η**

**Course Coverage:** STAT 700 and 701 divide roughly with definitions and properties for finite-sample statistics in the Fall (STAT 700), and large-sample limit theory in the Spring (STAT 701). The division is
not quite complete, because STAT 700 motivates many topics (Point Estimation, Confidence Intervals, identifiability) in terms of the Law of Large Numbers. The coverage in the Bickel & Doksum book for the
Fall is roughly Chapters 1-4 along with related reading in Casella & Berger for special topics.

**Grading:** There will be graded homework sets roughly every 1.5 weeks (7 or 8 altogether);
one in-class test, tenatively on Wed., Oct. 23; and an in-class Final Exam. The course grade will be
based 45% on homeworks, 20% on the in-class test, and 35% on the Exam.

**Homework will generally not be accepted late, and must be handed in as an uploaded pdf on ELMS.
(If you scan your handwritten papers or generate them using some other word-processor like Word or
LaTeX, then please convert them to pdf before uploading.)** Throughout the term, partial
problem set solutions will be posted to the ELMS course pages.

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)."

**The guideline for the course on Homeworks is that you may get hints from each other or from me, but
that you must write up your solutions completely by yourself, without copying any parts of solutions
from each other.**

The Fall 2024 Course Syllabus for Stat 700 is linked to this course web-page
(and also posted on the ELMS course pages).

**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 also through emails in the course-mail account.

For further information (updated throughout the term)
see the Important Dates below.

For auxiliary reading in several useful handouts that are described and linked below, click here.

**(I)** You can see
sample test problems for the 1st in-class test, along with the
Fall 2009 In-Class Test and a set of

sample problems for
the in-class final. Also see further sample Problems and Topics for the
Fall 2014 1st In-Class Test,

and sample Problems and Topics for the
Fall 2014 2nd In-Class Test.

**For the in-class test in Fall 2024, here are some Topics and Sample Problems for In-Class Test.**

**Here are lists of Important Topics and Sample Problems for the Fall 2022 Final Exam.**

**(II)** Old homework problem assignments in Casella and Berger from Fall 2014 can be found here. You can also see most of the solutions to these problems.

**(III)** I have a paper on the topic of
distributions related to the normal that are
or are not uniquely determined by their moments. (Hard-copy reprints available by request.) The paper uses many of the techniques we review in Chapters 1 and 3.

**(IV)** 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. Here is an additional handout specifically on the identifiability of 2-component normal mixtures.

**(V)** A paper for a talk I gave at the 2013 Federal Committee on Statistical Methodology includes the idea of following on those policies (in this case, for callback of nonrespondents in the American Community Survey) that are on the admissible frontier with respect to a specified set of loss functions. This idea is related to the discussion of admissibility in SDection 1.3.3 of Bickel and Doksum.

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

**(VII)** Handout on Cramer-Rao (Information) Inequality to supplement what we did in class and what is done in Bickel and Doksum's Section 3.4.2.

**(VIII)** Handout on EM Algorithm from STAT 705.

**(IX)** Lecture Notes from Stat 705 on Numerical Maximization of
Likelihoods.

**(X) 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 (X) below.

**(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.

**Homework:** Assignments, including any changes and hints, will continually be posted here. The most current

form of the assignment will be posted also on ELMS. The HW problems assigned in Fall 2022 are listed here.

**HW 1 due Wednesday Sept.4, 11:59pm (upload to ELMS)**

Readings to do during this time: Chapters 1, Sec.1.1 and Appendices A. 10-A.14 and B.7 of Bickel and Doksum.

**HW 2 due Wednesday Sept.18, 11:59pm (upload to ELMS)** Problems to solve include 1.1.3, 1.1.10, 1.5.5, 1.5.16

plus two more, given in LaTeX'ed pdf file here.

and Appendices A.14-A.15 of Bickel and Doksum, and Casella and Berger Sec.6.2.

**Note added on 9/13:** Problem 3 in HW2 has been modified. In the desired representation

h(E(g_{1}(X_{1})), E(g_{2}(X_{1})), ..., E(g_{k}(X_{1}))),
it is important that h is continuous, and there is no harm in also requiring it

to be smooth (continuously differentiable), but the functions g_{1}, g_{2}, ..., g_{k} are not required to be continuous.

**HW 3 due Wednesday Oct.2, 11:59pm (upload to ELMS)** Problems to solve include 1.2.4 in Bickel and Doksum,
plus five more, given in LaTeX'ed pdf file here.

**HW 4 due Friday Oct.18, 11:59pm (upload to ELMS)**
Problems to solve include 1.2.12, 1.2.14, 1.3.2, 1.3.6 (a),(b), and (e) in Bickel and Doksum,
plus one more, given in LaTeX'ed pdf file here.

**First Class: Mon., Aug. 26****No Class: Labor Day, Mon., Sept. 2****Mid-Term Exam: Wed., Oct. 23****Thanksgiving Recess: Wed., Nov. 27 through Sunday Dec.1****Last Class: Mon., Dec. 9****Review Session for Exam: Wed., December 11, 9-10:30 am (Room TBD)****Final Examination: Sat., December 14, 1:30-3:30 mm (PHYS 2124)**