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: If , , and is continuous in a neighborhood of , then \begin{eqnarray*} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where and ; here is the two-sided Strassen limit set on . 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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