Pointwise Convergence in Probability of General Smoothing Splines

Abstract

Establishing the convergence of splines can be cast as a variational problem which is amenable to a $\Gamma$-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, $n$, as ${\lambda }_n=n^{-p}$. Using standard theorems from the $\Gamma$-convergence literature, we prove that the general spline model is consistent in that estimators converge in a sense slightly weaker than weak convergence in probability for $p\le \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$.

References in corpus (2)

• Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems
• Local Asymptotics of P-splines