A central limit theorem for the Hellinger loss of Grenander‐type estimators

We consider Grenander‐type estimators for a monotone function , obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.