  We consider the estimation of the total number $N$ of species based on the
abundances of species that have been observed. We adopt a non parametric
approach where the true abundance distribution $p$ is only supposed to be
convex. From this assumption, we propose a definition for convex abundance
distributions. We use a least-squares estimate of the truncated version of $p$
under the convexity constraint. We deduce two estimators of the total number of
species, the asymptotic distribution of which are derived. We propose three
different procedures, including a bootstrap one, to obtain a confidence
interval for $N$. The performances of the estimators are assessed in a
simulation study and compared with competitors. The proposed method is
illustrated on several examples.
