MATH 3236 Spring 23

MATH 3236 Statistical Theory

Spring 2023

TR 15:30-16:45, Skiles 254

Office Hours: TR 18:00-19:00, Skiles 133b.

Professor Federico Bonetto

General Information

Description

This course is meant to provide an introduction to Statistical Theory for student in mathematics. The focus will be on the basic ideas of statistics and on the fundamental mathematical results of the theory as well as their application in real world situations. Thus, this class will contain formal definitions and rigorous proofs together with applications.

Prerequisites:

MATH 3235 Introduction to Probability

Course Goals and Learning Outcomes:

This class will make you familiar with the basics techniques and methods of statistical analysis and their application to real world problems. The focus will be on the construction of a consistent and rigorous probability setting to justify and guide the correct application of the statistical methods you will learn. The goal is to give you a strong and coherent mathematical foundation for a correct understanding and application of statistical analysis to real world problems and/or further development into more specialistic studies.

Course Requirements and Grading

There will be two midterm and one final. The midterms and final will be in the take home format. I will post a set of exercises meant to use the material learnt in class to analyze a concrete problem. You will have two or three days to work out the exercises. During this period I'll be available for clarifications or comments. Hopefully, in this way, these assignments will be an opportunity to learn. The date of the midterms are tentatively set for Tuesday February 28 and Tuesday April 4.

I will also assign homework from the textbook. This are mostly meant to check that you are following along with the class material.
The final grade will be based on HWs (10%), midterms (50%) and final (40%). I final score of 90% will grant an A while a final score of 80 will grant a B.
We will also discuss personal projects to be completed together with the final. To prepare a good project it is necessary to start as soon as possible. I can propose subjects for possible projects but I'd prefer if you find a problem involving statistical analysis that you are interested in and that can be analyzed with the tools you will learn in class.

Course Materials

Textbook

The class text book is:

M. H. De Groot, M. H. Schervish, Probability and Statistics, Addison Wesley, 4th edition

Web page

The weekly evolution of the class will be posted on the class web page, together with HW and extra class material. You can also check the we pages of my previous edition of this class: Sping 2020, Sping 2021.

Topic Outline:

Review: Large Samples Law of Large Numbers, Central Limit Theorem
Basics: Sample sets, Parametric statistical inference.
Estimstion: Point estimation, Confidence intervals.
Estimation Techniques: Method of Moments, Maximum Likelihood Estimation.
Asymptotic Results: Cramer-Rao Inequality, Asymptotic normality of the MLE.
Hypothesis Testing: Basics structure, Likelihood Ratio Tests, Chi-Squared Tests.
Applications: Introduction to regression, Analysis of Variance
Optional (time permiotting): Non-parametric methods

Class development:

First and Second week.

Material covered: Introduction and basic review of Probability Theory. Chapters 6.1, 6.2, 6.3 and 6.4.
Lecture notes: 1/17, 1/19

Exercises: Chapter 6.2 ex n. 3, 7, 15, Chapter 6.3 ex n. 4, 10, Chapter 6.4 ex. n. 2.

First HW collection: 1/26

Solution set for the first HW.

Third week.

Material covered: Chapters 7.1, 7.2.
Lecture notes: 1/24, 1/26

Exercises: Chapter 7.2: 3, 10.

Fourth week.

Material covered: Chapter 7.3 with Chapters 5.7 and 5.8.
Lecture notes: 1/31, 2/2

Exercises: Chapter 7.3: 7,8, 12, Chapter 5.7: 10, 19, Chapter 5.8: 5, 8

Second HW collection: 2/9

Fifth week.

Material covered: Chapter 7 section 7.4 and 7.5.
Lecture notes: 2/7, 2/9

Exercises: Chapter 7.4: 6, 12, 13, Chapter 7.5: 5, 9, 10

Solution set for the second HW.

Sixth week.

Material covered: Chapter 7 section 7.6 plus a proof of consistency of the ML estimator under mild regularity condition, from R.V. Hoggs, J.W. McKean and A.T. Craig, "Introduction to Mathematical Statistics" and a discussion of the asymptotic distribution of ML estimator from R.V. Hoggs and E.A. Tanis "Probability ans Statistical Inference".
Lecture notes: 2/14, 2/16

Exercises: Chapter 7.6: 9, 12, 23

Third HW collection: 2/23.

This are Spring 2020 and Spring 2021 midterm with solution for your practice.

As quoted in the Spring 2020 midterm, research on the relations among Pareto distribution, wealth distribution and economic inequality is active. Here a couple of example: On a Kinetic Model for a Simple Market Economy or Statistical Equilibrium Wealth Distribution in a Exchange Economy with Stochastic Preference . This is an area of research I'm quite interested in. I'll be more than happy to discuss with you if you find the question interestin

Seventh week.

Material covered: Chapter 8 section 8.8 and example and review for the midterm.
Lecture notes: 2/21, 2/23

Solution set for the third HW.

Eighth week.

Material covered: Chapter 8 section 8.1, 8.2
Lecture notes: 2/28, 3/2

Exercises: Chapter 8.1: 2, 9, Chapter 8.2: 4, 7, 10.

Solution set for the first midterm.

Ninth week.

Material covered: Multivariate Normal r.v. and Chapter 8 section 8.3
Lecture notes: 3/7, 3/9

Exercises: Chapter 8.3: 5, 7

Forth HW collection: 3/24.

Tenth week.

Material covered: Chapter 8 section 8.3 (cont.), 8.4
Lecture notes: 3/14, 3/16. Class recording: 3/16.

Exercises: Chapter 8.4: 3

The second midterm will be posted on Tuesday April 4th and due Thursday April 6th. It will cover up to Chapter 8.5 and concentrate on the material covered since Midterm 1. Here is some practice material from previous years: Spring 20, Spring 21.

Eleventh week.

Material covered: Chapter 8 section 8.5 and midterm review.
Lecture notes: 3/28, 3/30.

Exercises: Chapter 8.5: 1, 4, 7.

Fifth HW collection: 4/11.

Solution set for the forth HW.

Twelfth week.

Material covered: Chapter 9 section 9.1 and midterm 2 solution.
Lecture notes: 4/4, 4/6.

Exercises: Chapter 9.1: 2, 4, 8, 10, 19

Solution set for the second midterm.

Thirteenth week.

Material covered: Chapter 9 section 9.1 (continued).
Lecture notes: 4/11, 4/13.

Solution set for the fifth HW.

Fourteenth week.

Material covered: Chapter 9 section 9.13 and 9.5.
Lecture notes: 4/18, 4/20.

Fifteenth week.

Material covered: Class review
Lecture notes: 4/25.

The final will be posted on Tuesday 4/25 around noon and will be due on Thursday 4/27 before 5:30pm. It will cover from Chapter 1 to Chapters 5.6, 5.7, 5.8, 6.1, 6.2, 6.3, 7.1 to 7.6, 8.1 to 8.5, 9.1, 9.3, 9.5 and the extra material posted on the sixth week. Here is some practice material from previous years: Spring 20, Spring-21.