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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 f(t0)>0f(t_0) > 0, f(t0)<0f'(t_0) < 0, and ff' is continuous in a neighborhood of t0t_0, 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 MsupgGTg=(3/4)1/3 M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3} and Tg\mboxargmaxu{g(u)u2} T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} ; here G{\cal G} is the two-sided Strassen limit set on RR. 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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