A law of the iterated logarithm for Grenander's estimator

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Dumbgen, Lutz¹  Wellner, Jon A.²  Wolff, Malcolm² 

[1] Univ Bern, Inst Math Stat & Actuarial Sci, Alpeneggstr 22, CH-3012 Bern, Switzerland

[2] Univ Washington, Dept Stat, Seattle, WA 98195 USA

关键词: Grenander;    Monotone density;    Law of iterated logarithm;    Limit set;    Strassen;    Switching;    Strong invariance theorem;    Limsup;    Liminf;    Local empirical process;   

DOI  :  10.1016/j.spa.2016.04.012

来源: Elsevier

【 摘 要 】

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f (t(0)) > 0, f' (t(0)) < 0, and f' is continuous in a neighborhood of t(0), then lim sup(n ->infinity) (n/2log log n)(1/3) (<(f)over cap>(n)(t(0)) - f(t(0))) = vertical bar f(t(0)) f'(t(0))/2 vertical bar(1/3) 2m almost surely where M equivalent to sup(g epsilon G) T-g = (3/4)(1/3) and T-g equivalent to argmax(u){g(u) - u(2)}; 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]. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

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