Introduction

The approximation and the estimation of nonparametric functions by projections on an orthonormal basis of functions are useful in data analysis.

Series Estimators

This book presents series estimators defined by projections on bases of functions, extending the estimators of densities to mixture models, deconvolution, and inverse problems. They are applicable to semi-parametric and nonparametric models for regressions, hazard functions, and diffusions.

Estimation and Convergence

These estimators are calculated in the Hilbert spaces with respect to the distribution function of the regressors, and their optimal rates of convergence are proved. Their mean square errors depend on the size of the basis, which is estimated via cross-validation.

Wavelet Estimators

Wavelet estimators are defined and studied within the same models. The choice of the basis, with appropriate parametrizations, and their estimation improve existing methods and enable applications to a wide class of models.

Rates of Convergence and Applications

The rates of convergence of the series estimators are the best among all nonparametric estimators, with significant improvements in multidimensional models. The book also discusses original methods developed for estimation in deconvolution and inverse problems. Additionally, the asymptotic properties of test statistics based on these estimators are established.