ESTIMATING THE VARIANCE IN NONPARAMETRIC REGRESSION - WHAT IS A REASONABLE CHOICE

The exact mean-squared error (MSE) of estimators of the variance in no nparametric regression based on quadratic forms is investigated. In pa rticular, two classes of estimators are compared: Hall, Kay and Titter ington's optimal difference-based estimators and a class of ordinary d ifference-based estimators which generalize methods proposed by Rice a nd Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of th e integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference-based estimat ors are more appropriate for estimating the variance, because they con trol the bias much better and hence have a much better overall perform ance. It is also demonstrated that Rice's estimator does not always be have well. Data-driven guidelines are given to select the estimator wi th the smallest MSE.