Nonparametic estimation and testing

Objective

This course offers an introduction to nonparametric estimation. We will cover standard nonparametric methods for density estimation (kernel estimators, projection onto a Fourier or wavelet basis) and for regression (local polynomial estimators, the Nadaraya–Watson estimator, projection-based estimators). We will conclude with a statistical analysis of a generative model: generative adversarial networks (GANs).

Throughout, we will emphasize non-asymptotic risk guarantees for the estimators under study, depending on the regularity of the object of interest (measured in a Hölder or Sobolev space).

Planning

The following topics will be covered:

• Kernel and projection estimators of a density. Cross-validation. Rates of convergence.

• Nonparametric estimation of the regression function. Local polynomial estimators; projection estimators (Fourier bases, wavelet bases). Rates of convergence.

• Approximation theory: Sobolev, Hölder, and Besov spaces.

• Generative models via generative adversarial networks (GANs).

References

L. Devroye: A Course in Density Estimation. Birkhauser, Boston, 1987.

E. Giné, R. Nickl: Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge University Press, 2015
A.Nemirovski: Topics in non-parametric statistics. Ecole d'Eté de Probabilités de Saint-Flour XXVIII – 1998. Lecture Notes in Mathematics, v.1738. Springer, 2000.
A.B.Tsybakov: Introduction to Nonparametric Estimation. Springer, New York, 2009.
L. Wasserman: All of Nonparametric Statistics. Springer, New York, 2006.