标题： 对数凹密度及其分布函数的最大似然估计：基本性质和一致收敛性 显示英文标题

标题： Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

摘要： 我们研究了对数凹概率密度及其分布函数和风险函数的非参数最大似然估计。通过两种特征化方法推导出这些估计量的一些一般性质。结果表明，密度函数和风险函数的估计量在紧区间上关于一致范数的收敛速度至少为$(\log(n)/n)^{1/3}$，通常为$(\log(n)/n)^{2/5}$；而在某些正则性假设下，经验分布函数与估计分布函数之间的差异以速率$o_{\mathrm{p}}(n^{-1/2})$消失。 显示英文摘要

摘要： We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\log(n)/n)^{1/3}$ and typically $(\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions.