Math 728: Statistical theory.
Spring 2017
TR   01:00 -02:15 PM   SNOW 302

January 17 -- May 4.
See this link for the KU calendar year.

Terry Soo, Snow 610
Office hours: Tuesday 2:30 - 3:30.  Friday 2-3.

Course description.
In this course we will cover core topics in point estimation, hypothesis testing, and Bayesian statistics.  Highlights include the Cramer-Rao bound, the Rao-Blackwell theorem, the Lehmann-Scheffe theorem and the Neyman-Pearson lemma.

Prerequisites: Math 727. 
In particular, students should be comfortable with conditional expectation, the law of large numbers, and the central limit theorem.      
Proofs will be an important part of the course.   Students should be comfortable with reading and writing proofs. 
This is a graduate course.
See the review sheet: math728rev.pdf

Textbook and lecture notes. 
No textbook is required.  We will follow course lecture notes which can be found here, which is the course website for 2016.  Previous homework assignments and exams are also available there.  Lecture notes will be updated and supplemented, and posted at this webpage. 

Other suitable references are: 

Introduction to Mathematical Statistics, Hogg, McKean, and Craig.  (Used last year)

Statistical Inference, Casella and Berger (Used in previous years)

Mathematical Statisitics, Shao
Theory of Point Estimation, Lehmann and Casella
Testing Statistical Hypothesis, Lehmann and Romano

This course will also have a minor computing component.   We will have the chance to use the free statistical software R.  It can be downloaded here.   A short introduction to R can be found here.   Knowledge of R will not be required on examinations.  However, there may be a few R homework assignments.    

Qualifying exam.
This course will help students who are preparing for the Probability and Statistics qualifying examination, and requires a high level of mathematical maturity.

Subject to revision.

Homework: 30%. (Either weekly or biweekly)
Midterm 1:  10%.  February 14  M1    sol
Midterm 2:  10%.  March 14  sol
Midterm 3:  10%.  April 20   sol
Final examination: 40%. May 12: 1.30 -- 4.00 PM  registrar information

Notes.  Also see last year's notes, here

Maximum likelihood estimatation

Consistency of mle
Fisher information and the Cramer-Rao Bound
Central limit theorem for mle
Sufficient statistics
Change of variables
Conditional Expectation
Complete statistics
Laplace transforms
Exponential class
Higher dimensions
Introduction to R, sample code
The EM algorithm
Intro BayesI
Intro BayesII
Undergraduate hypothesis testing and confidence intervals
Introduction to Hypothesis testing and the LRT
Neyman-Pearson lemma for best tests
Uniformly most powerful tests and monotone likelihood ratios


Homework 1: Due January 31
Updated Question 1; it is a better question now and also added the missing assumption of independence.
For Question 2, you do not need to know anything fancy about the distribution of a sum of independent uniforms, the reason being is that you only need to know the cdf of the sum up to value 1, and no more no less.  Use induction.
For Question 5b, there was a missing term in the event, this is now corrected.
For Question 5e, there was a typo, it should have said Zk = Finverse(Vk); this has been corrected in the pdf file.
For Question 7, the pdf for the multivariate normal is provided.

I encourage you to typeset your solutions using LaTeX.  For you reference here is a LaTex file that can be used to generate this homework . 
latexfile.  (Texfiles will not be updated)

Homework 2: Due February 7
Comments:  In Question 1, there should be a negative sign was missing in one of the equations that defined symmetric.  This has been corrected.
There was a typo in the solution to Exercise 5. This did not effect the final answer and has been corrected. See the remark in the solutions for details.

Homework 3: Due February 16

Homework 4: Due February 28

Homework 5: Due March 7

Homework 6: Due Monday March 13 3PM
In Exercise 2, the definition of T is the sample sum; this was left out in the previous version, and is now corrected.

Homework 7:  Due April 11

Homework R: Due April 25
Comments: There should be a negative sign in the exponent in Ex4a.

Homework 9: Due April 18

Last HW: Due May 4
In the first question, the assumption of symmetry in M is not necessary, all that you need is that it has zero median.