## 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 language | English |
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Article number | 110182 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 213 |

DOIs | |

Publication status | Published - 2024 |

## Keywords

- Bias–variance trade-off
- Mean absolute deviation
- Minimax estimation
- Nonparametric estimation