# Pointwise convergence in probability of general smoothing splines

*Matthew Thorpe* () and
*Adam Johansen* ()

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Matthew Thorpe: Carnegie Mellon University

*Annals of the Institute of Statistical Mathematics*, 2018, vol. 70, issue 4, 717-744

**Abstract:**
Abstract Establishing the convergence of splines can be cast as a variational problem which is amenable to a $$\Gamma $$ Γ -convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, n, as $$\lambda _n=n^{-p}$$ λ n = n - p . Using standard theorems from the $$\Gamma $$ Γ -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 $$p\le \frac{1}{2}$$ p ≤ 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 $$p>\frac{1}{2}$$ p > 1 2 .

**Keywords:** Variational methods; $$\Gamma $$ Γ -convergence; Pointwise convergence; General spline model; Nonparametric smoothing (search for similar items in EconPapers)

**Date:** 2018

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