Least Squares Consistent Estimates for Arbitrary Regression Functions Over an Abstract Space

Abstract

In this article, we propose a new approach to sieve estimation for a general regression function when the dimension of the finite dimensional subspaces is a random quantity depending on the values of the observations.

The technique is introduced with the help of a simulation study on a functional linear model under extremely mild assumptions.

A sketch of the proof concerning the main statements is then given in the more general case when the regression function is not necessarily linear.

Keywords:

• Functional estimation
• Least squares technique
• Sieves estimation

Mathematics Subject Classification:

• Primary 62G08
• Secondary 62M86, 62J02