Class
meetings: Tuesday
and Thursday 3:30  4:45, in Hanes 125.
Prerequisites:
Elementary probability
(see
below for more details) and at least one semester of
real analysis.
Instructor:
Andrew B. Nobel,
Department of Statistics and Operations Research
Office: Hanes
308 Phone: 9621352.
Office
Hours: Mondays 1:30pm2:30pm
Teaching
assistant:
Guan Yu
Office: Hanes B40 Email: guanyu@live.unc.edu
Grader's
Office
Hours: Mondays and Wednesdays 11am12am
Homework
policy:
Homework
problems will be assigned periodically throughout the
semester. Each homework assignment will be graded:
late/missed homeworks will receive a grade of zero. Students
are welcome to discuss the homework problems with other
members of the class, but should prepare their final answers
on their own. If you have any questions concerning the
grading of homework, please speak first with the TA. If you
are absent from class when an assignment is returned, you
can get your homework from the TA during their office hours.
Exams:
There will be
one inclass midterm exam, and a comprehensive final
examination. Each will be closed book, closed notes.
Midterm
1

Tuesday
9 October 
Final

The default time is that listed in the course directory. The exam will be in class. 
Grading: Course
grades will be calculated as follows:
Homework 
15% 
Midterm 1 
35% 
Final 
50% 
Required Text: The
lectures will (roughly) follow Chapters 69 of the book "Statistical
Inference", Second Edition, by G. Casella and R.L. Berger,
Duxbury, 2002.
Prerequisites:
Students should be familiar with basic undergraduate
probability, including the topics listed below.
1. Basic properties of
probablities, random variables. Probability mass
functions and probability density functions. Cumulative
distribution functions.
2.
Binomial, geometric, hypergeometric, Poisson, and negative
binomial distributions.
3.
Gaussian, uniform, exponential, double exponential, gamma,
chisquared and beta distributions. F and t
distributions.
4. Joint probability mass/density functions, independence.
4.
Expected values, moments, variance and covariance.
5.
Distributions of functions of a random variable. General
change of variables formula.
Prerequisites
can
be found in Casella and Berger: 1.16, 2.13, 3.13, 4.13,
4.56, 5.12. (Note that these sections also contain
some material that is not prerequisite for the course.)
Students may also wish to consult the textbook "A
First Course in Probability", by Sheldon Ross, or the
``bootcamp'' lecture notes.
Syllabus: The
course is intended to introduce students to some of the
basic ideas and techniques of (nonasymptotic) statistical
inference from a theoretical point of view. The course is
not measuretheoretic, but we will be mathematically
rigorous whenever possible. The following is an overview of
the topics covered.
1. Order statistics,
Stirling's formula, moment generating functions, convex
functions and their basic properties
2. Basic inequalities from analysis: Jensen, Holder,
CauchySchwartz, association inequalities for
monotone functions
3. Basic probability inequalities: Markov, Chebyshev,
and the Chernoff bound
4.
Elements of decision theory: admissability, Bayes and minimax
procedures.
5.
Principles of data reduction: sufficiency and minimal
sufficiency (from Chapter 6 of CB)
6.
Introduction to point estimation: maximum likelihood, method
of moments (from Chapter 7 of CB)
7.
Introduction to hypothesis testing: likelihood ratio tests,
NeymanPearson theorem (from Chapter 8 of CB)
8. Introduction to confidence intervals: inverting test statistics, pivotal quantities (from Chapter 9 of CB)
9.
Exponential inequalities for sums of independent random
variables: Hoeffding, Bennett, and Bernstein
10. Elementary concentration: EfronStein inequality,
martingale differences, and McDiarmid's inequality
11. Conditional expectations as L_2 projections: definitions
and basic properties.
12. Expectations and variancecovariance matrices for random
vectors
13. The multivariate Gaussian distribution: definition and
basic properties, conditional distributions, independence of
linear and quadratic forms
14. Representation of correlated Gaussian random variables,
elementary Gaussian comparison results (Slepian's Lemma)
"Mathematical Statistics",
Second Edition, by P.J. Bickel and K.A. Doksum, Prentice Hall,
2001.
"Theory of Point Estimation'', by E.L. Lehmann, Wadsworth,
1991.
"Mathematical Statistics",
Second Edition, by J. Shao, Springer, 2003.
Other Texts of Potential Interest:
"Elements
of Information Theory", by T. Cover and J. Thomas, Wiley,
1991.
"Combinatorial
Methods
in Density Estimation", by L. Devroye and G. Lugosi, Springer,
2001.
(More on probability inequalities. See also the nice
lecture notes on concentration inequalities by Boucheron,
Lugosi and Massart, available from Lugosi's web page.)
"Principles
of Mathematical Analysis", Third Edition, by W. Rudin, McGraw
Hill. (This book is a good reference for real analysis.)
"Linear
Algebra and its Applications", Third Edition, by G. Strang,
Saunders, 1988. (This book gives a good basic overview of
linear algebra.)
"Asymptotic
Statistics",
by A.W. van der Vaart, Cambridge University Press, 2000.
(A good advanced text on theoretical statistics.)
"Linear Statistical Inference
and its Applications", by C.R. Rao, Wiley, 1973.
"The CauchySchwartz Master Class'', J.M. Steele, Cambridge,
2004. (A very well written and insightful book on basic
inequalities, which can be read with relatively little
background. Contains many interesting problems, with
solutions.)