English
 
User Manual Privacy Policy Disclaimer Contact us
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT
  Variational multiscale nonparametric regression: Smooth functions.

Grasmair, M., Li, H., & Munk, A. (2018). Variational multiscale nonparametric regression: Smooth functions. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 54(2), 1058-1097. doi: 10.1214/17-AIHP832.

Item is

Basic

show hide
Item Permalink: http://hdl.handle.net/21.11116/0000-0001-6E8D-6 Version Permalink: http://hdl.handle.net/21.11116/0000-0003-52B1-8
Genre: Journal Article

Files

show Files
hide Files
:
2597697.pdf (Preprint), 6MB
Name:
2597697.pdf
Description:
-
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
Technical Metadata:
Copyright Date:
-
Copyright Info:
-
License:
-

Locators

show

Creators

show
hide
 Creators:
Grasmair, M., Author
Li, H.1, Author              
Munk, A.1, Author              
Affiliations:
1Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society, ou_1113580              

Content

show
hide
Free keywords: Nonparametric regression; Adaptation; Convergence rates; Minimax optimality; Multiresolution norm; Approximate source conditions
 Abstract: For the problem of nonparametric regression of smooth functions, we reconsider and analyze a constrained variational approach, which we call the MultIscale Nemirovski-Dantzig (MIND) estimator. This can be viewed as a multiscale extension of the Dantzig selector (Ann. Statist. 35 (2009) 2313-2351) based on early ideas of Nemirovski (J. Comput. System Sci. 23 (1986) 111). MIND minimizes a homogeneous Sobolev norm under the constraint that the multiresolution norm of the residual is bounded by a universal threshold. The main contribution of this paper is the derivation of convergence rates of MIND with respect to L-q-loss, 1 <= q <= infinity, both almost surely and in expectation. To this end, we introduce the method of approximate source conditions. For a one-dimensional signal, these can be translated into approximation properties of B-splines. A remarkable consequence is that MIND attains almost minimax optimal rates simultaneously for a large range of Sobolev and Besov classes, which provides certain adaptation. Complimentary to the asymptotic analysis, we examine the finite sample performance of MIND by numerical simulations. A MATLAB package is available online.

Details

show
hide
Language(s): eng - English
 Dates: 2015-12-032018-03-272018-05
 Publication Status: Published in print
 Pages: -
 Publishing info: -
 Table of Contents: -
 Rev. Method: Peer
 Identifiers: DOI: 10.1214/17-AIHP832
 Degree: -

Event

show

Legal Case

show

Project information

show

Source 1

show
hide
Title: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Source Genre: Journal
 Creator(s):
Affiliations:
Publ. Info: -
Pages: - Volume / Issue: 54 (2) Sequence Number: - Start / End Page: 1058 - 1097 Identifier: -