  In this paper we develop rate--optimal estimation procedures in the problem
of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from
independent observations. The density is assumed to be defined on $R^d$, $d\geq
1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the
minimax approach and construct rate--optimal estimators in the case of integer
$p\geq 2$. We demonstrate that, depending on parameters of Nikolskii's class
and the norm index $p$, the risk asymptotics ranges from inconsistency to
$\sqrt{n}$--estimation. The results in this paper complement the minimax lower
bounds derived in the companion paper \cite{gl20}.
