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Lower bounds for the trade-off between bias and mean absolute deviation

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  • Derumigny, Alexis
  • Schmidt-Hieber, Johannes

Abstract

In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function f in a class of β-Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions f in the Hölder class. It is shown that any estimator with worst-case bias ≲n−β/(2β+1)≕ψn must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ψn. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

Suggested Citation

  • Derumigny, Alexis & Schmidt-Hieber, Johannes, 2024. "Lower bounds for the trade-off between bias and mean absolute deviation," Statistics & Probability Letters, Elsevier, vol. 213(C).
    • Handle: RePEc:eee:stapro:v:213:y:2024:i:c:s0167715224001512
    DOI: 10.1016/j.spl.2024.110182
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    References listed on IDEAS

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    1. Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers 29/13, Institute for Fiscal Studies.
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