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

Alexis Derumigny, Johannes Schmidt-Hieber*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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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)≕ψ n must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ψ n. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

Original languageEnglish
Article number110182
Number of pages6
JournalStatistics and Probability Letters
Volume213
DOIs
Publication statusPublished - Oct 2024

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