Grenander估计量的重对数律
A law of the iterated logarithm for Grenander's estimator
摘要：
本文证明了单调递减密度的Grenander估计的重对数律：若$f(t_0) > 0$，$f'(t_0) < 0$，$f'$在$t_0$的邻域内连续，则\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*}几乎必然其中$ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3}$，$ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $ ;这里${\cal G}$是$R$上的双边Strassen极限集.证明依赖于当地的经验过程，Groeneboom的开关关系，和斯特拉森的极限集类似于布朗运动的分布特性的法律的重对数。
英文摘要：
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 \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 $ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3}$ and $ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $; here ${\cal G}$ 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.
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