Instructor: Marten Wegkamp
Description: We discuss recent theoretical developments
in nonparametric regression, density estimation and classification.
We introduce some basic empirical process theory and related tricks. Emphasis
will be put on universal consistency and model selection. There is no required
textbook, although I will use various sources: monographs by Devroye and
Lugosi (2001), Van de Geer (2000), combined with articles, and ongoing
research.
Below you find an outline of the course.
Prerequisites: Stat330/Stat600 or equivalent.
Syllabus:
I. Empirical process theory
A. Uniform laws of large
numbers.
B. Rates of convergence.
C. VC classes.
II. Universal bandwidth selection
A. Density estimation: Devroye
and Lugosi (1996,1997), Wegkamp (1999).
B. Regression estimation:
Hengartner and Wegkamp (2001).
III. Nonparametric regression
A. Least squares regression:
consistency, rates of convergence.
B. Universal consistency:
Kohler.
C. Monotone regression:
Groeneboom.
D. Model selection: Baraud
(1999) and Wegkamp (2001)
IV. Nonparametric classification
A. Stone�s result: universal
consistency.
B. Structural risk minimization:
penalties related to VC-dimension.
C. Random penalties: Koltchinskii
and Panchenko (2001) and Lugosi and Wegkamp (2002).
V. Special models: data reduction
A. Partial linear models.
B. Direct estimation of
the index coefficient in a single index model (Hristache, Juditsky and
Spokoiny (2001).