Statistics 607b, Spring 2005
Inequalities for probability and statistics
Instructor: Mr. D. Pollard.
Time: MW 1:00-2:15
Office hours: TBA
Place: 24 HH Rm. 107
A guided tour of some inequalities useful in statistical
and probabilistic problems. The course will be broken into
independent segments, each treating a specific method and
an illustrative application. Acquaintance with probability
at the 600 level helpful for some segments. Possible
topics: convexity arguments; tail bounds for martingales
and independent summands; metric entropy and maximal
inequalities; VC dimension and combinatorial methods;
distances between probability measures; majorizing measures
and generic chaining; isoperimetric inequalities;
concentration inequalities; Gaussian processes.
Applications to: statistical inference; asymptotic theory;
minimax rates of convergence; machine learning; complexity.
Grading
Homework plus one more substantial project.
References
See the reference list for some books and original papers.
I will also prepare handouts, with other material cited, for some topics.
Intended topics
I cannot possibly cover everything in the following list properly in one semester. Topics near the end will probably be covered only briefly or omitted altogether.
- Tails of the normal distribution: classical bounds;
Mill's ratio; conditional distributions; LIL.
- Convexity:
log moment generating functions; Hoeffding's inequality; Bennett's
inequality; large deviation bounds;
Orlicz norms; subgaussian distributions.
- Metric entropy: chaining inequality for processes via
Orlicz norms; Gaussian processes.
- Distances between measures: total variation; Hellinger; Kullback-Leibler; Fano's lemma; Assouad's inequality; minimax rates of convergence; maximum likelihood in one dimension via Hellinger distance.
- Covariance inequalities: FKG; application to 2-dimensional Ising model.
- Combinatorial bounds: Vapnik-Cervonenkis and beyond.
- Concentration inequalities: martingale method; Talagrand's method for product measures; tensorization methods.
- Poisson and Binomial: coupling bound; Le Cam-Hodges method; Chen-Stein method; unimodality; majorization.
- Berry-Esseen bound: via Fourier inversion method; Stein's method.
- Tusnady's inequality: coupling of empirical and Gaussian processes.
- Majorizing measures: the "generic chaining".
- Sudakov and Fernique inequalities for Gaussian processes; Borell's isopermetric inequality.