Quadratic functional estimation from observations with multiplicative measurement error
We consider the nonparametric estimation of the value of a quadratic functional evaluated at the density of a strictly positive random variable X based on an iid. sample from an observation Y of X corrupted by an independent multiplicative error U. Quadratic functionals of the density covered are th...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
23 July 2025
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| In: |
Annals of the Institute of Statistical Mathematics
Year: 2025, Pages: 1-41 |
| ISSN: | 1572-9052 |
| DOI: | 10.1007/s10463-025-00936-x |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s10463-025-00936-x
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| Author Notes: | Fabienne Comte, Jan Johannes, Bianca Neubert |
| Summary: | We consider the nonparametric estimation of the value of a quadratic functional evaluated at the density of a strictly positive random variable X based on an iid. sample from an observation Y of X corrupted by an independent multiplicative error U. Quadratic functionals of the density covered are the $${\mathbb{L}^{2} }$$-norm of the density and its derivatives or the survival function. We construct a fully data-driven estimator when the error density is known. The plug-in estimator is based on a density estimation combining the estimation of the Mellin transform of the Y density and a spectral cut-off regularized inversion of the Mellin transform of the error density. The main issue is the data-driven choice of the cut-off parameter using a Goldenshluger-Lepski-method. We discuss conditions under which the fully data-driven estimator attains oracle-rates up to logarithmic deteriorations. We compute convergence rates under classical smoothness assumptions and illustrate them by a simulation study. |
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| Item Description: | Veröffentlicht: 23. Juli 2025 Gesehen am 19.11.2025 |
| Physical Description: | Online Resource |
| ISSN: | 1572-9052 |
| DOI: | 10.1007/s10463-025-00936-x |