  This paper deals with the classical problem of density estimation on the real
line. Most of the existing papers devoted to minimax properties assume that the
support of the underlying density is bounded and known. But this assumption may
be very difficult to handle in practice. In this work, we show that, exactly as
a curse of dimensionality exists when the data lie in $\R^d$, there exists a
curse of support as well when the support of the density is infinite. As for
the dimensionality problem where the rates of convergence deteriorate when the
dimension grows, the minimax rates of convergence may deteriorate as well when
the support becomes infinite. This problem is not purely theoretical since the
simulations show that the support-dependent methods are really affected in
practice by the size of the density support, or by the weight of the density
tail. We propose a method based on a biorthogonal wavelet thresholding rule
that is adaptive with respect to the nature of the support and the regularity
of the signal, but that is also robust in practice to this curse of support.
The threshold, that is proposed here, is very accurately calibrated so that the
gap between optimal theoretical and practical tuning parameters is almost
filled.
