Naive Penalized Spline Estimators of Derivatives Achieve Optimal Rates of Convergence

TL;DR

This paper demonstrates that differentiating penalized spline estimators of a mean function yields optimal convergence rates for derivative estimation, simplifying the process while maintaining statistical efficiency.

Contribution

It shows that straightforward differentiation of penalized spline estimators achieves the optimal L2 convergence rate for derivatives, providing a simple yet effective method.

Findings

Differentiated penalized spline estimators attain optimal L2 convergence rates.

The approach simplifies derivative estimation without sacrificing accuracy.

Theoretical analysis confirms the optimality of the method.

Abstract

This paper studies the asymptotic behavior of penalized spline estimates of derivatives. In particular, we show that simply differentiating the penalized spline estimator of the mean regression function itself to estimate the corresponding derivative achieves the optimal L2 rate of convergence.