@techreport{2c27effa0c654add81cd2d5e4c6b04a9,
title = "Lower bounds for the trade-off between bias and mean absolute deviation",
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 $\beta$-H\{"}older smooth functions. Let 'worst-case' refer to the supremum over all functions $f$ in the H\{"}older class. It is shown that any estimator with worst-case bias $\lesssim n^{-\beta/(2\beta+1)}=: \psi_n$ must necessarily also have a worst-case mean absolute deviation that is lower bounded by $\gtrsim \psi_n.$ To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation. ",
keywords = "math.ST, stat.TH, 62C20, 62G05, 62C05",
author = "Alexis Derumigny and Johannes Schmidt-Hieber",
note = "This is an extended version of Section 7 of arXiv:2006.00278v3. The material has been removed from later versions of arXiv:2006.00278",
year = "2023",
month = mar,
day = "21",
doi = "10.48550/arXiv.2303.11706",
language = "English",
series = "Statistics and Probability Letters, Volume",
publisher = "ArXiv.org",
type = "WorkingPaper",
institution = "ArXiv.org",
}