With the recent advances in powerful computing and the availability of massive sets of data, the tools of statistics, data science, and analytics have become indispensable in the applied sciences and in industry. This course focuses on the mathematical underpinnings that provide the foundations to modern day data analysis and statistics.
The contents of the course may depend in part on the background, preparation, and interest of the students, thus making the list of topics below somewhat flexible. Some topics may be omitted due to time constraints.
- Parametric models, linear models, variable selection, the lasso and matrix completion
- Estimation, criteria and construction of estimators, maximum likelihood, asymptotics
- Non parametric models, empirical distribution function, bootstrap
- Hypothesis testing, multiple hypotheses testing, family wise error, false discovery rate
- Density and regression estimation, regularization and smoothing
- Classification, discriminant analysis, Vapnik Chervonenkis (VC) dimension
Course Prerequisite: Students should have at least one good course in probability, and some basic statistics. It will be assumed that students are familiar with the first five chapters of the course text, All of Statistics: A concise course in Statistical Inference, by Larry Wasserman. Students should review these chapters and study any material new to them before starting the course.
It is also strongly recommended that students read Chapter 6 of the textbook, which consists mostly of material that is covered in first year statistics courses (e.g., confidence intervals, testing hypotheses).
Instructor: Larry Goldstein
Structure and Evaluation
Course participation, 25%, Midterm Exam (Friday July 28th, 30%), Final Exam, 45%
Course Text
All of Statistics: A concise course in Statistical Inference, by Larry Wasserman.
Additional References
- Mathematical Statistics: Basic Ideas and Selected Topics, by Peter Bickel and Kjell Doksum
- A Course in Large Sample Theory, by Thomas Ferguson
- Statistical Inference, by Casella and Berger
- Theoretical Statistics: Topics for a Core Course, by Keener
- High Dimensional Statistics: A non-asymptotic view, by Martin Wainwright
- The Convex Geometry of Linear Inverse Problems, V. Chandrasekaran, B. Recht, P. Parrilo, A. Willsky
- The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy, N. Heckman
- Midterm Solutions
- Notes on Linear Models, Information Inequality, Estimating Equations, Multivariate Analysis, Model Selection, Multiple Testing
Assignments:
A Selection of Works by the Instructor:
Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula
Goldstein, L., Nourdin, I. and Peccati, G.
Annals of Applied Probability (2017), vol 27, pp. 1-47
[http://arxiv.org/abs/1411.6265]
M-estimation in a diffusion model with application to biosensor transdermal blood alcohol monitoring
Allayioti, M., Bartroff, J., Goldstein, L., Luczak, S. and Rosen, G.
[https://arxiv.org/abs/2002.05335]
Gaussian random field approximation via Stein’s method with applications to wide random neural networks
Balasubramanian, K., Goldstein, L., Ross, N. and Salim, A.
[https://arxiv.org/abs/2306.16308]
Other Links of Interest
- Anonymous Course Feedback Form
- Netflix $1,000,000 prize
- Why most published research findings are false.
- Benford’s law, Simple Explanation, Basic Theory of Benford’s law (for scale invariance, see Example 4.19 on page 42)
- Formula for one dimensional normal density
- Box Muller Transformation
- Perugia Web Cam, Piazza IV novembre
- Frontone Cinema all’Aperto