Pointwise Convergence in Probability of General Smoothing Splines

2015/06/28 by Matthew Thorpe, Thorpe, Matthew, Adam M. Johansen +1
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1506.08450

arxiv created 2017/03/11 · arxiv updated 2017/03/14

Abstract

Establishing the convergence of splines can be cast as a variational problem which is amenable to a Γ-convergence approach. We consider the case in which the regularization coefficient scales with the number of observations, n, as λ_n=n^-p. Using standard theorems from the 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≤ (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>(1)/(2).

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