Mini-course "Nonparametric Curve Estimation"

Professor Enno Mammen (Heidelberg University)
Academic Superviser of the Laboratory of Stochastic Analysis and its Applications (HSE)

The course took place in HSE, September 15-20, Shabolovka st.26, Moscow, Russia.
This course introduces into the basic techniques of nonparametric smoothing. In particular, we will discuss kernel smoothing, sieve estimators and smoothing splines. The course discusses applications of the methods in semi parametric models and inverse problems that e.g. arise in nonparametric models with instrumental variables. A further topic is structured nonparametric models where several nonparametric functions enter into a model specification. 
The aim of the course is an understanding of the theoretical mathematical background of these models and statistical approaches. 
Lecture 1. Nonparametric kernel density estimation. Motivation of kernel density estimators; asymptotic theory: limit distributions and rates of convergence; smoothing parameter selection; multivariate density estimation. 

Lecture 2. Bootstrap of kernel density estimation. Bootstrap and smooth bootstrap; asymptotic theory of the bootstrap: consistency for pointwise estimation and for supremum statistics; bootstrap confidence bands. 

Lecture 3. Nonparametric kernel regression estimation. Nadaraya-Watson estimator; local linear estimation and local polynomial estimation; asymptotic theory; boundary problems; confidence intervals and bands. 

Lecture 4. Nonparametric tests. Nonparametric tests based on L_2 and sup-norms; asymptotic theory; asymptotic power; bootstrap tests; wild bootstrap. 

Lecture 5. Well-posed inverse problems. Additive models; empirical integral equation; plug-in estimators of integral equations; backfitting and smooth backfitting estimators; asymptotic theory.

Lecture 6. Ill-posed inverse problems. Sieve estimation; optimal convergence rates; nonparametric instrumental regression; background from functional analysis.