Adaptive and optimal estimation under shape and smoothness constraints

We consider the problem of estimating the density $f$ of a real valued random variable. We present a fully adaptive procedure based on a wavelet series expansion of $f$. We study the $\mathbb{L}^1$ risk of our estimator when $f$ satisfies a mild smoothness condition on $\mathbb{R}$, i.e $f$ belongs to a (weak) Besov class. Its tails are assumed to be $s$-monotone and may decay as a power law. We show the importance of considering the negative resolutions of the wavelet expansion to optimally estimate a fat tailed density. We also reveal new minimax rates. In particular, we explain when the shape constraint improves the estimation rate of a smooth density.