The limit distribution of the $L_{\mathrm{\infty }}$-error of Grenander-type estimators

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n\mathrm{/}\log n{)}^{-1\mathrm{/}3}$ and the limiting distribution to be Gumbel.