| lecturer |
A.W. van der Vaart
(at VU) |
| credits | 8 |
| period | Spring 2011 |
| schedule |
Wednesday 14.00-16.30, in weeks 6-21.
First meeting: February 9. No meeting: Wednesday March 23
and Wednesday 20 April. |
| location |
VU University; weeks 6 - 11 and 13 - 20: WN-M639,
weeks 12 and 21: WN-121 (WN=science building),
Science Building Vrije Universiteit |
| exam | Written, 1 June, 14-17 hours, M 623. |
| Retake, Wednesday 17 August, 14-17 hours. |
| See here
for example questions. |
| registration | Registration for the course
via mastermath.nl is necessary.
Grades will be registered via mastermath. |
| contents |
The empirical measure of a set of random variables is the discrete
random measure that puts a point mass of size one divided by the
sample size at each of the random variables. The expectation of some
function under this measure is just an average, and under appropriate
integrability this average will satisfy a law of large numbers (LLN)
and, after centering and scaling, a central limit theorem
(CLT). Empirical process theory studies these objects for many
functions jointly, and is concerned with the LLN or CLT uniformly in
classes of functions, as well as inequalities that measure the size of
suprema of these objects over classes of functions. The empirical
distribution function and the classical empirical process on the line
are very special examples, for which the uniform LLN and CLT were
obtained by Glivenko-Cantelli in the 1930s and Donsker in the
1940/50s, respectively. The Kolmogorov-Smirnov statistic for
goodness-of-fit and its approximation by the maximum of a Brownian
bridge process is one important application of these classical
results. The general theory of empirical processess is more recent,
and is based on Vapnik-Cervonenkis combinatorial theory and
Kolmogorov entropy.
This theory has many applications in statistics. In this
course we shall focus on its use to derive rates of estimation of
nonparametric statistical procedures. In the terminology of computer
science this is called statistical learning theory. For instance, one
obtains a sample of instances (realizations of variables in some
measurable space), each being classified as a 0 or 1, and one wants to
build a procedure that can classify a future instance as a 0 or 1.
Empirical risk minimization, support vector machines, and kernel
learners are all methods to solve this problem, and can be studied
using empirical process theory.
|
| literature |
Chapters from Weak Convergence and Empirical Processes
by Van der Vaart and Wellner (2nd edition).
Excerpts will appear on this site. |
|
|
| required knowledge |
Measure-theoretic probability. |