Quadratic functional estimation from observations with multiplicative measurement error
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909286212304896 |
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| author | Neubert, Bianca Comte, Fabienne Johannes, Jan |
| author_facet | Neubert, Bianca Comte, Fabienne Johannes, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic functional estimation from observations with multiplicative measurement error Neubert, Bianca Comte, Fabienne Johannes, Jan Statistics Theory Primary 62G05, Secondary 62G07 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. |
| title | Quadratic functional estimation from observations with multiplicative measurement error |
| topic | Statistics Theory Primary 62G05, Secondary 62G07 |
| url | https://arxiv.org/abs/2408.06862 |