Class
meetings: Tuesday and
Thursday 3:30 - 4:45, in Peabody 217.
Prerequisites:
Elementary
probability and at least one semester of real analysis
Instructor:
Andrew B. Nobel,
Department of Statistics and O.R.
Office: Smith 302
Phone:
962-1352.
Office
Hours: After class, and
by appointment.
Teaching
assistant: Sungkyu Jung
Office: TBA Phone: TBA Email:
sungkyu@email.unc.edu
Grader's Office Hours: TBA
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
in-class midterm exam, and a comprehensive final examination.
Each will be closed book, closed notes.
|
Midterm
1 |
TBA. |
|
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% |
Syllabus:
The course is meant to
present and familiarize students with some of the basic tools of
theoretical
and applied statistics. The course is not measure-theoretic, but we
will be
rigorous whenever possible. The following is an overview of the topics
covered
in the course. (A more complete syllabus can be found on the
course web page.)
1.
Random variables, distribution functions and densities.
2. Univariate distribution theory: gamma, beta, chi-squared, normal, t and F distributions, convolutions, change of variables.
3.
Expected values, moments, conditional expectations.
4.
Characteristic functions and moment generating functions.
5.
Convexity and basic probability inequalities: Jensen, Holder,
Minkowski, Markov, Chernoff, and Hoeffding.
7.
Asymptotics: Types of convergence, the weak law of large numbers,
the central limit theorem, Slutsky's theorem, the delta-method, O_p and
o_p
notation.
8.
Multivariate distribution theory: Multivariate CDF's, expected
values and covariances of random vectors, multivariate characteristic
functions,
multivariate normal distributions, conditioning and the Fisher-Cochran
theorem.
Recommended Texts: There is no
required
text for the course. Course material and background can be found in a
number of
good books, which are listed below. The book of Casella and
Berger has
the best overall coverage of the material in the course, though it does
not
cover the multivariate normal distribution in great detail. For
the
latter the book of Mardia, Kent and Bibby is useful, though we only
cover some
of the material appearing in Chapters 1-3. The appendix of
MK&B gives
a nice overview of the matrix algebra necessary for the course.
For
students unfamiliar with matrix algebra, or needing a fast review, I
suggest
that you look over this appendix during the first half of the
semester.
Please contact me if you have specific questions about where to look
for a
particular result or subject.
*
"Statistical
Inference", Second Edition, by G. Casella and R.L. Berger, Duxbury,
2002.
* "Multivariate Analysis", by K.V. Mardia, J.T. Kent and J.M. Bibby,
Academic Press, 1979.
``The Cauchy-Schwartz 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 lots of nice problems, with solutions.)
"A First
Course in Probability", Sixth Edition, by S. Ross,
Prentice Hall, 2002. (Good reference for basic probability, the
CDF
method, and basic results about univariate distributions.)
"Mathematical
Statistics", Second Edition, by P.J. Bickel and K.A.
Doksum, Prentice Hall, 2001.
"Elements of
Information Theory", by T. Cover and J. Thomas,
Wiley, 1991. (This book gives a complete coverage of the material on
entropy
that we survey in the course.)
"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. (An good advanced text on theoretical
statistics.)
"Probability
and Measure", Third Edition, by P. Billingsley,
Wiley, 1995.
"Linear
Statistical Inference and its Applications", by C.R. Rao,
Wiley, 1973.