Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5130017 | Stochastic Processes and their Applications | 2016 | 11 Pages |
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, fâ²(t0)<0, and fâ² is continuous in a neighborhood of t0, then blalim supnââ(n2loglogn)1/3(fÌn(t0)âf(t0))=|f(t0)fâ²(t0)/2|1/32M almost surely where Mâ¡supgâGTg=(3/4)1/3andTgâ¡argmaxu{g(u)âu2}; here G is the two-sided Strassen limit set on R. 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; see Strassen [26].
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Lutz Dümbgen, Jon A. Wellner, Malcolm Wolff,