Pointwise Convergence in Probability of General Smoothing Splines

Matthew Thorpe, Adam Johansen

Research output: Contribution to journalArticlepeer-review


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≤12. 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>12.
Original languageEnglish
Pages (from-to)717-744
JournalAnnals of the Institute of Statistical Mathematics
Issue number4
Early online date4 Apr 2017
Publication statusPublished - Aug 2018


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