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

We consider Grenander type estimators for a monotone function $λ:[0,1]\to\mathbb{R}$, 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 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 $λ$.

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