Pointwise Convergence in Probability of General Smoothing Splines

TL;DR

This paper proves the pointwise convergence in probability of general smoothing splines when the regularization parameter scales with the number of observations, using a $\Gamma$-convergence approach, and identifies the sharpness of the convergence rate.

Contribution

It introduces a $\Gamma$-convergence framework to analyze spline consistency and establishes the optimal rate of convergence for the regularization parameter scaling.

Findings

Spline estimators converge in probability for p ≤ 1/2

The convergence rate p ≤ 1/2 is shown to be sharp

The approach differs from Hilbert scale methods for strong convergence

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 $p\leq \frac{1}{2}$. Without further assumptions we show this rate is sharp. This differs from rates for strong convergence using Hilbert scales where one can often choose $p>\frac{1}{2}$.