SAMPLE PROBLEMS FOR IN-CLASS FINAL CAN BE FOUND HERE.
SAMPLE FOR IN-CLASS TESTS CAN BE FOUND HERE.
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 begin by studying topics related to
(finite-sample) hypothesis testing and confidence regions,
but for the
rest of the semester 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 Estimatiion
Equation
solutions. As time permits, we will also something about
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 Peter Bickel and Kjell Doksum, Mathematical
Statistics, vol.I, 2nd ed., Pearson Prentice Hall, 2007.
Some problems
and alternative explanations will be taken from:
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:
Bickel and Doksum: Chapters 4--6.
Rohatgi and Saleh: Sections 9.5--9.6, 10.2, Chapter 11 omitting Sec. 11.4.
Course Grading: there will be assigned and graded homework
due approximately every 1.5 weeks (probably 7 in all). Homework
will count 45% toward the course grade, Test(s) [in-class plus
take-home] will count 35%, and final
exam will count 20%.
Click link here for syllabus, and here for the
Homework Assignments. Selected problem
solutions are given here.
Office hours: are Monday 1-2 and Thursday 11-12. I
will be available very often except on Tuesdays, but please send an e-mail
or arrange with me in
class for an office appointment.
The topic coverage of the in-class Mid-term is as follows: TBA
There will be a second test, a Take-home, in the first week of
May. There will also be an in-class final.
(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 the 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.
Several typos have now been corrected (particularly in formulas (9)-(11)).
(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 May 10, 2010.