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Adaptation Bounds for Confidence Bands under Self-Similarity

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Abstract

We derive bounds on the scope for a confidence band to adapt to the unknown regularity of a nonparametric function that is observed with noise, such as a regression function or density, under the self-similarity condition proposed by Gine and Nickl (2010). We find that adaptation can only be achieved up to a term that depends on the choice of the constant used to define self-similarity, and that this term becomes arbitrarily large for conservative choices of the self-similarity constant. We construct a confidence band that achieves this bound, up to a constant term that does not depend on the self-similarity constant. Our results suggest that care must be taken in choosing and interpreting the constant that defines self-similarity, since the dependence of adaptive confidence bands on this constant cannot be made to disappear asymptotically.

Suggested Citation

  • Timothy B. Armstrong, 2018. "Adaptation Bounds for Confidence Bands under Self-Similarity," Cowles Foundation Discussion Papers 2146, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:2146
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d21/d2146.pdf
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    1. Timothy B. Armstrong, 2018. "Adaptation Bounds for Confidence Bands under Self-Similarity," Cowles Foundation Discussion Papers 2146, Cowles Foundation for Research in Economics, Yale University.
    2. Susanne M Schennach, 2020. "A Bias Bound Approach to Non-parametric Inference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 87(5), pages 2439-2472.
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    Cited by:

    1. Timothy B. Armstrong, 2018. "Adaptation Bounds for Confidence Bands under Self-Similarity," Cowles Foundation Discussion Papers 2146R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2019.

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    More about this item

    Keywords

    Adaptation; Nonparametric inference; Self-similarity;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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