Pré-Publication, Document De Travail Année : 2025

PROJECTION ESTIMATION OF A QUADRATIC FUNCTIONAL FROM INDIRECT OBSERVATIONS

Résumé

We consider the convolution model: $Y = X + \varepsilon$, where $X$ and $\varepsilon$ are independent random variables.We aim to estimate the quadratic functional $\int_\mathbb{R} f^2(x)dx$, where $ f$ is the unknown density of the signal $X$ from $n$ observations of $Y$. We introduce a new projection estimator by expanding $f$ in the Hermite basis. Convergence rates for $f$ within the Sobolev-Hermite ball are provided. We also develop an adaptive procedure inspired by the method of Goldenshluger and Lepski (2011) to select the appropriate projection space. An oracle inequality is proved for the adaptive estimator when the noise density is known. Additionally, we show that our method can be extended to the case of an unknown noise density. Numerical experiments are conducted to illustrate the effectiveness of our methodology.

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Dates et versions

hal-04559411 , version 1 (25-04-2024)
hal-04559411 , version 2 (05-09-2024)
hal-04559411 , version 3 (04-04-2025)
hal-04559411 , version 4 (26-11-2025)

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  • HAL Id : hal-04559411 , version 4

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Ousmane Sacko. PROJECTION ESTIMATION OF A QUADRATIC FUNCTIONAL FROM INDIRECT OBSERVATIONS. 2025. ⟨hal-04559411v4⟩
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