Adaptive deconvolution on the nonnegative real line

Abstract : In this paper we consider the problem of adaptive density or survival function estimation in an additive model defined by $ Z = X + Y$ with $X$ independent of $Y$, when both random variables are nonnegative. We want to recover the distribution of $X$ (density or survival function) through $n$ observations of $Z$, assuming that the distribution of $Y$ is known. This issue can be seen as the classical statistical problem of deconvolution which has been tackled in many cases using Fourier-type approaches. Nonetheless, in the present case the random variables have the particularity to be $\mathbb{R}^+$ supported. Knowing that, we propose a new angle of attack by building a projection estimator with an appropriate Laguerre basis. We present upper bounds on the mean squared integrated risk of our density and survival function estimators. We then describe a nonparametric adaptive strategy for selecting a relevant projection space. The procedures are illustrated with simulated data and compared to the performances of more classical deconvolution setting using a Fourier approach.
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https://hal.archives-ouvertes.fr/hal-01076927
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Submitted on : Wednesday, November 16, 2016 - 4:16:37 PM
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Gwennaëlle Mabon. Adaptive deconvolution on the nonnegative real line. 2016. ⟨hal-01076927v3⟩

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