A law of the iterated logarithm for Grenander’s estimator

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• 作者：Lutz Dü ; mbgen^a ; Jon A. Wellner^b ; ^{jaw@stat.washington.edu" class="auth_mail" title="E-mail the corresponding author} ; Malcolm Wolff^b
• 关键词：60F15 ; 60F17 ; 62E20 ; 62F12 ; 62G20
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：126
• 期：12
• 页码：3854-3864
• 全文大小：255 K

文摘

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$f\left(t_0\right)>0$, f^′(t₀)<0$f^{\prime }\left(t_0\right)<0$, and f^′$f^{\prime }$ is continuous in a neighborhood of t₀$t_0$, then almost surely where here G$\mathcal{G}$ is the two-sided Strassen limit set on R$\mathbb{R}$. 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].