Description
In this talk, we construct a nonparametric estimator of the density $f:\mathbb R_+ \rightarrow \mathbb R_+$ of a positive random variable $X$ based on an i.i.d. sample $(Y_1, \dots, Y_n)$ of
\begin{equation}Y=X\cdot U,
\end{equation} where $U$ is a second positive random variable independent of $X$. More precisely, we consider the case where the distribution of $U$ is unknown but an i.i.d. sample $(\widetilde U_1, \dots, \widetilde U_m)$ of the error random variable $U$ is given.
Based on the estimation of the Mellin transforms of $Y$ and $U$, and a spectral cut-off regularisation of the inverse Mellin transform, we propose a fully data-driven density estimator where the choice of the spectral cut-off parameter is dealt by a model selection approach. We demonstrate the reasonable performance of our estimator using a Monte-Carlo simulation.
