MATH 3236 Statistical Theory
TR 15:30-16:45, Remote synchronous
Office Hours: TR 18:00-19:00, Remote synchronous.
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 Modality: The course will be remote synchronous. Class will be held using Bluejeans (my favourite) or Teams. We will experiment with this at the beginning of class.
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 several days to work out the exercises. During this period I'll be available to help and direct your work. Hopefully, in this way, these assignments will be an opportunity to learn. The date of the midterms are tentatively set for Thursday February 25 and Thursday April 1. .
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%).
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 differential equation you are interested in and that can be analyzed with the tools you will learn in class.
The class text book is:
M. H. De Groot, M. H. Schervish, Probability and Statistics, Addison Wesley, 4th edition
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.
- 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
First and Second week.
Material covered: Introduction and basic review of Probability Theory. Chapters 6.1, 6.2, 6.3 and 6.4.
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: 1/28
Solution set for the first HW.
Material covered: Chapters 7.1, 7.2
Exercises: Chapter 7.2: 3, 10.
Material covered: Chapter 7.3 with Chapters 5.7 and 5.8.
Exercises: Chapter 7.3: 7,8, 12, Chapter 5.7: 10, 19, Chapter 5.8: 5, 8
Second HW collection: 2/11
Material covered: Chapter 7 section 7.4 and 7.5.
Exercises: Chapter 7.4: 6, 12, 13, Chapter 7.5: 5, 9, 10
Solution set for the second HW.
Sixth and Seventh week.
Material covered: Chapter 7 section 7.6 and 6.6 from R.V. Hoggs and E.A. Tanis "Probability ans Statistical Inference". For interested people this optional material contain a proof of consistency of the MLE estimator under mild regularity condition, from R.V. Hoggs, J.W. McKean and A.T. Craig, "Introduction to Mathematical Statistics".
Exercises: Chapter 7.6: 9, 12, 23
This is last year midterm with solution for your practice.
As quoted in the practice text, 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
Material covered: Chapter 8 section 8.1, 8.2 and 8.5
Exercises: Chapter 8.1: 2, 9, Chapter 8.2: 4, 7, 10, Chapter 8.5: 1, 4, 7
Material covered: Chapter 8 section 8.3, 8.4 and 8.7
Exercises: Chapter 8.3: 5, 7 Chapter 8.4: 3, Chapter 8.5: 1, 4, 7, Chapter 8.7: 6, 11
Here is the solution set for the third HW and also last year second midterm as preparation material for the coming midterm.
Material covered: Chapter 9 section 9.1.
Exercises: Chapter 9.1: 2, 4, 8, 10, 19
Solution set for the second midterm.