ABSTRACT
This chapter presents pivotal results for the classical case of directly observed data. It is explicitly assumed that observations are neither modified nor missing. The chapter is intended to overview basics of orthonormal series approximation, to present a universal method of orthonormal series estimation of nonparametric curves and to explain adaptive estimation of the probability density and regression function for the case of complete data. It explains the problem of nonparametric density estimation. Here the nonparametric E-estimator, which will be used for all considered in the book problems, is introduced and explained. The chapter is devoted to the classical problem of nonparametric regression estimation. It is explained how the E-estimator, proposed for the density model, can be used for the regression model. In many applied settings with modified and/or missing data, a special type of a nonparametric regression, called a Bernoulli (binary) regression, plays a key role.
