title: A law of the iterated logarithm for Grenander's estimator.
abstract: In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: Iff(t0) > 0,f'(t0)<0, andf'is continuous in a neighborhood oft0, 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.
authors.affiliation: University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, CH-3012 Bern.
authors.forename: Lutz
authors.initials: L
authors.lastname: Dümbgen
authors.affiliation: Department of Statistics, University of Washington, Seattle, WA 98195-4322.
authors.forename: Jon A
authors.initials: JA
authors.lastname: Wellner
authors.affiliation: Department of Statistics, University of Washington, Seattle, WA 98195-4322.
authors.forename: Malcolm
authors.initials: M
authors.lastname: Wolff
keywords: Grenander; Strassen; law of iterated logarithm; liminf; limit set; limsup; local empirical process; monotone density; strong invariance theorem; switching