This course introduces mathematical statistics at a theoretical
graduate level, using tools of advanced calculus and basic analysis.
The objectives are to treat diverse statistically interesting models for
data in a conceptually unified way; to define mathematical
properties which
good procedures of statistical inference should have; and to prove that
some common procedures have them.
In the Spring term, (Stat 701), we will emphasize large-sample theory results,
especially the large-sample properties of
Maximum Likelihood Estimators and
(Generalized) Likelihood Ratio Statistics and, more generally, of Estimating
Equation
solutions. We will discuss related statistical computing topics,
especially Bootstrap methods, plus Jackknife estimators (mostly,
of variance) and
the EM algorithm. We will also cover some topics
in nonparametric (`rank') statistics which might be used
with smaller or
moderate sized samples but which make most sense with larger samples.
Prerequisite: Stat 700 or equivalent. You should be comfortable
(after review) with joint densities, (multivariate, Jacobian)
changes of
variable, moment generating functions, and conditional expectation; and also
familiar with the definitions of
convergence in distribution, in probability,
and convergence with probability 1.
Texts: required
Jun Shao, Mathematical Statistics, 2nd ed., Springer, 2003.
recommended
Peter Bickel and Kjell Doksum, Mathematical
Statistics, vol.I, 2nd ed., Pearson Prentice Hall, 2007.
V. Rohatgi and A.K. Saleh,
An Introduction to Probability and Statistics, 2nd ed., Wiley.
Some material
on large-sample theory will be taken from:
Thomas S. Ferguson, A Course in Large Sample Theory, Chapman & Hall, 1996.
Approximate Stat 701 course coverage: Large-sample theory topics from Shao text:
Ch.1 (Sec.1.5), Ch.2 (Sec.2.5), Ch.3
(Secs. 3.1.4, 3.2 and 3.5), Ch.4 (Secs. 4.3.3, 4.4 and 4.5), Ch. 5 (all) ,
Ch.6 (Secs. 6.4.2
and 6.5), Ch.7 (Secs. 7.3 and 7.4).
Corresponding topics in Bickel and Doksum: Chapters 4--6.
Course Grading: there will be assigned and graded homework
due approximately every 1.5 weeks
(probably 7 in all). Homework
will count 40% toward the course grade, Test(s) [1 in-class 20% plus
1
take-home 15%], and final exam will count 25%.
Course Policies:
(I) Homeworks will
generally be due on a Friday, and should be submitted as hard-copy.
If you
will be off-campus then you may submit these electronically, but in that
case you are
expected to hand in a hard-copy at the following class meeting.
(II) Late Homeworks will be
accepted up to one class meeting later than the due date,
but you will lose 20% credit for late submission.
(II) Makeup tests will
not be given except for written medically excused illness.
(III) Ideas may be shared
among students in working on homeworks, but it is essential
that you
write up the solutions explaining the steps and methods fully in your own
words.
For Take-home Test(s), you may not share ideas with each other and
must not accept help from
anyone other than the course instructor.
Click link here for syllabus, and here for
the Homework Assignments, including HW4.
Office hours: are Monday 11-12 and Wed 4-5. I
will be available very often except not on Tuesdays
and Thursdays, but please
send an e-mail or arrange with me in
class for an office appointment.
(I). Handout on Prediction
intervals in (simple) linear regression in connection with
Prediction Intervals topic in Bickel & Doksum, Sec. 4.8.
(II). Summary of calculations in R comparing three
methods for creating (one-sided)
confidence
intervals for binomial proportions in moderate sized samples.
(III). Handout on Chi-square
multinomial goodness of fit test.
(IV). 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.
(V). Handout on the 2x2
table asymptotics covered in a 2009 class concerning different
sampling
designs and asymptotic distribution theory for the log
odds ratio.
(VI). Handout on Wald, Score
and LR statistics covered in class April 10 and 13, 2009.
(VII). 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.
(VIII). 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.
(IX). A handout of topics and sample problems
given for the March 31, 2014, in-class test, can be found
here.
(X). An updated (in Spring 2014)
directory of sample problems for the Final Exam can be found here.