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

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