Statistics 9720 Mathematical Statistics II Winter 2007

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Statistics 9720

Mathematical Statistics II

Winter 2007


Instructor


Marco A. R. Ferreira

Office



134
-
O Middlebush Hall (884
-
8568)

Email



ferreiram@missouri.edu

Hours



Tuesday and Thursday 2
-
3pm and by appointment


Text



Ferguson,
A Course in Large Samp
le Theory


References


Billingsley,
Probability and Measure

Cramer,
Mathematical Methods of Statistics

Lehmann,
Elements of Large
-
Sample Theory

Lehmann and Casella,
Theory of Point Estimation

Prakasa Rao, B. L. S.,
Asymptotic Theory of Statistical Inferenc
e

Rao, C. R.,
Linear Statistical Inference and Its Applications
,
second edition (especially chapters 1
-
3, 5, 6)

Schervish,
Theory of Statistics

Serfling,
Approximation Theorems of Mathematical Statistics





Other references

Little and Rubin,
Statistical A
nalysis with Missing data

McCullogh and Nelder,
Generalized Linear Models (Second
Edition)

Efron and Tibshirani,
An Introduction to the Bootstrap


Grading


Homework (30%), two midterms and final (70%)



Students with disabilities
: If you have special need
s as addressed by the Americans
with Disabilities Act (ADA) and need assistance, please notify the Office of Disability
Services, A048 Brady Commons, 882
-
4696 or the course instructor immediately.
Reasonable efforts will be made to accommodate your specia
l needs.


Honesty
: Academic honesty is fundamental to the activities and principles of a
university. All members of the academic community must be confident that each
person’s work has been responsibly and honorably acquired, developed, and presented.
A
ny effort to gain an advantage not given to all students is dishonest whether or not the
effort is successful. The academic community regards academic dishonesty as an
extremely serious matter, with serious consequences that range from probation to
expuls
ion. When in doubt about plagiarism, paraphrasing, quoting, or collaboration,
consult the course instructor.



Syllabus


I.

Preliminaries


1.

Overview of Lebesgue integral, absolute continuity, densities

2.

Convergence in probability, laws of large numbers

3.

Con
vergence in distribution

4.

Continuity theorem for characteristic functions (no proof)

5.

Central limit theorems including Lindeberg and Liapunov conditions (no proof)

6.

Cramer
-
Wold theorem, Multivariate central limit theorem

7.

Transformations and delta method

8.

Order

statistics and asymptotic distribution of quantiles


II.

Asymptotic methods of inference


1.

Asymptotic normality of multinomial vectors, asymptotic distribution of
goodness
-
of
-
fit chi
-
square statistic with and without estimated parameters

2.

Fisher information

and Cramer
-
Rao lower bound

3.

Maximum likelihood theory: consistency and asymptotic normality

4.

Method of scoring

5.

Asymptotic normality of Bayes posterior mode and posterior distribution.

6.

Asymptotic distribution of the likelihood ratio test, Rao’s test and Wald
’s test.


III.

Other topics


1.

EM algorithm

2.

Some theory of jackknife and bootstrap

3.

Introduction to generalized linear models, inference

4.

Topics at discretion of instructor