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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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Hauptverfasser: Comte, Fabienne (VerfasserIn) , Johannes, Jan (VerfasserIn) , Neubert, Bianca (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: 23 July 2025
In: Annals of the Institute of Statistical Mathematics
Year: 2025, Pages: 1-41
ISSN:1572-9052
DOI:10.1007/s10463-025-00936-x
Online-Zugang:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s10463-025-00936-x
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Verfasserangaben:Fabienne Comte, Jan Johannes, Bianca Neubert

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520 |a 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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