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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.11706 |
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| _version_ | 1866916294424526848 |
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| author | Derumigny, Alexis Schmidt-Hieber, Johannes |
| author_facet | Derumigny, Alexis Schmidt-Hieber, Johannes |
| contents | 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 $\lesssim n^{-β/(2β+1)}=: ψ_n$ must necessarily also have a worst-case mean absolute deviation that is lower bounded by $\gtrsim ψ_n.$ To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11706 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the trade-off between bias and mean absolute deviation Derumigny, Alexis Schmidt-Hieber, Johannes Statistics Theory 62C20, 62G05, 62C05 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 $\lesssim n^{-β/(2β+1)}=: ψ_n$ must necessarily also have a worst-case mean absolute deviation that is lower bounded by $\gtrsim ψ_n.$ To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation. |
| title | Lower bounds for the trade-off between bias and mean absolute deviation |
| topic | Statistics Theory 62C20, 62G05, 62C05 |
| url | https://arxiv.org/abs/2303.11706 |