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A law of the iterated logarithm for Grenander's estimator
Lutz Dümbgen1, Jon A Wellner2, Malcolm Wolff2
1University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, CH-3012 Bern.
Abstract:
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0) > 0, f'(t0) < 0, and f' is continuous in a neighborhood of t0, then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion.
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