Consistency for the least squares estimator in nonparametric regression

Consistency for the Least Squares Estimator in Non-parametric Regression

We shall study the general regression model Y = g 0 (X) + ", where X and " are independent. The available information about g 0 can be expressed by g 0 2 G for some class G. As an estimator of g 0 we choose the least squares estimator. We shall give necessary and suucient conditions for consistency of this estimator in terms of (basically) geometric properties of G. Our main tool will be the th...

Nonparametric regression estimation using penalized least squares

We present multivariate penalized least squares regression estimates. We use Vapnik{ Chervonenkis theory and bounds on the covering numbers to analyze convergence of the estimates. We show strong consistency of the truncated versions of the estimates without any conditions on the underlying distribution.

Level Robust Methods Based on the Least Squares Regression Estimator

Consistency for Least Squares Regression Estimators with Infinite Variance Data

The least squares estimators are discussed for the linear regression model with random predictors. Both predictors and errors may have infinite variance. Under the condition that the predictors are in a stable domain of attraction, we determine necessary and sufficient conditions for weak consistency of the least squares estimators in the simple linear model. The conditions vary, depending on w...

Local nonlinear least squares: Using parametric information in nonparametric regression

We introduce a new nonparametric regression estimator that uses prior information on regression shape in the form of a parametric model. In e!ect, we nonparametrically encompass the parametric model. We obtain estimates of the regression function and its derivatives along with local parameter estimates that can be interpreted from within the parametric model. We establish the uniform consistenc...