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Pointwise convergence in probability of general smoothing splines
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Thorpe, Matthew and Johansen, Adam M. (2018) Pointwise convergence in probability of general smoothing splines. Annals of the Institute of Statistical Mathematics, 70 (4). pp. 717-744. doi:10.1007/s10463-017-0609-x ISSN 0020-3157.
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Official URL: https://doi.org/10.1007/s10463-017-0609-x
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 λn = 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
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||||
Library of Congress Subject Headings (LCSH): | Spline theory, Convergence, Nonparametric statistics, Probabilities | ||||||||
Journal or Publication Title: | Annals of the Institute of Statistical Mathematics | ||||||||
Publisher: | Springer | ||||||||
ISSN: | 0020-3157 | ||||||||
Official Date: | April 2018 | ||||||||
Dates: |
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Volume: | 70 | ||||||||
Number: | 4 | ||||||||
Page Range: | pp. 717-744 | ||||||||
DOI: | 10.1007/s10463-017-0609-x | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 20 April 2017 | ||||||||
Date of first compliant Open Access: | 4 April 2018 | ||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) |
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