A new approach to least-squares estimation, with applications. (English) Zbl 0625.62046

van de Geer, Sara

Ann. Stat. 15, 587-602 (1987).

The regression model \(y=g(x)+\epsilon\) and least-squares estimation are studied in a general context. For an estimator of the unknown g to be statistically meaningful, it should at least be consistent in some sense. In the least-squares context, the most natural requirement is \(L^ 2\)- consistency.
In this paper the author shows that entropy conditions on a rescaled and truncated version of the class of regression functions \({\mathcal G}\) imply strong \(L^ 2\)-consistency of the least-squares estimator. A result from empirical process theory is used to prove this. He deals with a few examples, such as (non)linear regression and isotonic regression. Some nonparametric regression estimators can also be considered as least- squares estimators, or modifications thereof (for instance penalized least squares).

Reviewer: U.B.Paik

MSC:

62J02	General nonlinear regression
62J05	Linear regression; mixed models
62G05	Nonparametric estimation
60B10	Convergence of probability measures

Keywords:

strong \(L^ 2\)-consistency; empirical measure; uniform convergence; least-squares estimation; entropy conditions; empirical process; isotonic regression; nonparametric regression estimators; penalized least squares