  We consider Grenander type estimators for a monotone function
$\lambda:[0,1]\to\mathbb{R}$, obtained as the slope of a concave (convex)
estimate of the primitive of $\lambda$. Our main result is a central limit
theorem for the Hellinger loss, which applies to statistical models that
satisfy the setup in Durot (2007). This includes estimation of a monotone
density, for which the limiting variance of the Hellinger loss turns out to be
independent of $\lambda$.
