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

Abstract

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

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