Density estimation for nonnegative random variables
Résumé
We propose a new type of non parametric density estimators fitted to nonnegative random variables. The estimators are constructed using kernels which are densities of empirical means of $m$ i.i.d. nonnegative random variables with expectation 1. The value $m^{-1/2}$ plays the role of the bandwidth. We study the pointwise Mean Square Error and a weighted global Mean Integrated Square Error and propose adaptive estimators for both local and global points of view. The risks of the adaptive estimators satisfy oracle inequalities. A noteworthy result is that the adaptive rates are in correspondence with the smoothness properties of the unknown density as a function on $(0,+\infty)$. Pointwise adaptive estimators are illustrated on simulated data.
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