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A Law of the Iterated Logarithm for Grenander's Estimator

Monday, October 24, 2016 - 2:30pm
THO 325

Speaker: Jon Wellner (University of Washington)

Abstract: I will discuss the following law of the iterated logarithm for the Grenander estimator (or MLE) \$\widehat{f}_n\$ 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 \begin{eqnarray*} \phantom{bla} \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 $$ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3} \ \ \ \mbox{and} \ \ \ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} ; $$ here \${\cal G}\$ is the two-sided Strassen limit set on \$\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. The constant in the limsup is related to the tail behavior of Chernoff's distribution, the law of \$\mbox{argmax} \{W(u) - u^2\}\$ where \$W\$ is two-sided Brownian motion starting at \$0\$. I will also briefly discuss several open problems connected to the asymptotic distribution of the mode estimator \$M(\widehat{f}_n)\$ where \$\widehat{f}_n\$ is the MLE of a log-concave density \$f\$ on \$\mathbb{R}\$ with \$(-\log f)^{\prime \prime } (m) > 0\$.


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