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

Alexis Derumigny and Johannes Schmidt-Hieber

Statistics & Probability Letters, 2024, vol. 213, issue C

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.

Keywords: Bias–variance trade-off; Mean absolute deviation; Minimax estimation; Nonparametric estimation (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spl.2024.110182

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