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Statistical inference for the optimal approximating model
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  • Published: 22 February 2012

Statistical inference for the optimal approximating model

  • Angelika Rohde1 &
  • Lutz Dümbgen2 

Probability Theory and Related Fields volume 155, pages 839–865 (2013)Cite this article

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  • 3 Citations

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Abstract

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001–1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ 2-distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.

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Author information

Authors and Affiliations

  1. Department Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Angelika Rohde

  2. Institut für Mathematische Statistik und Versicherungslehre, Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Lutz Dümbgen

Authors
  1. Angelika Rohde
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  2. Lutz Dümbgen
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Corresponding author

Correspondence to Angelika Rohde.

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Cite this article

Rohde, A., Dümbgen, L. Statistical inference for the optimal approximating model. Probab. Theory Relat. Fields 155, 839–865 (2013). https://doi.org/10.1007/s00440-012-0414-7

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  • Received: 28 January 2010

  • Revised: 06 January 2012

  • Published: 22 February 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0414-7

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Keywords

  • Adaptivity
  • Confidence regions
  • Coupling
  • Exponential inequality
  • Model selection
  • Multiscale inference
  • Risk optimality

Mathematics Subject Classification (2000)

  • 62G15
  • 62G20
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