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The limit distribution of the \(L_{\infty}\)-error of Grenander-type estimators. (English) Zbl 1257.62017

Summary: 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 \((n/\log n)^{-1/3}\) and the limiting distribution to be Gumbel.

MSC:

62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62G07 Density estimation
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