Abstract: We consider the problem of estimating the derivatives of a regression function by the corresponding derivatives of regression splines. Unlike kernel smoothers, these spline derivative estimators do not have boundary problems. In addition, they have simple expressions and are easy to compute. In this paper, we study the local asymptotic properties of these derivative estimators. Under regularity conditions, the asymptotic bias and variance of these estimators are derived, and asymptotic normality is established. Furthermore, we extend the results to random designs and heteroscedastic errors.
Key words and phrases: Asymptotic bias, asymptotic normality, asymptotic variance, Bernoulli polynomial, B-splines, derivative estimators, heteroscedastic errors, random designs, regression splines.