Grenander functionals and Cauchy's formula
Grenander functionals and Cauchy's formula
复制标题
格林纳德泛函和柯西公式
DOI：
10.1111/sjos.12449
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发表时间：
2019
影响因子：
1
通讯作者：
P. Groeneboom
中科院分区：
文献类型：
--
作者：
P. Groeneboom
设f^n是递减密度的非参数极大似然估计。Grenander将其描述为经验分布函数的最小凹优数的左连续斜率。对于来自均匀分布的样本，Grenander估计量到均匀密度的L2-距离的渐近分布在Groeneboom和Pyke的一篇文章中通过使用条件泊松和伽马随机变量的Grenander估计量的表示导出。Groeneboom和Lopuhaä在一篇文章中也使用了这种表示来证明Sparre Andersen关于Grenander估计的跳跃次数的中心极限结果。在这里，我们将其扩展到证明Grenander估计到均匀密度的L2距离的主要结果，并且还证明了熵泛函的类似渐近正态性结果。柯西公式和鞍点方法是我们发展的主要工具。
Let f^n be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left‐continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the L2‐distance of the Grenander estimator to the uniform density was derived in an article by Groeneboom and Pyke by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in an article by Groeneboom and Lopuhaä to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of the main result on the L2‐distance of the Grenander estimator to the uniform density and also prove a similar asymptotic normality results for the entropy functional. Cauchy's formula and saddle point methods are the main tools in our development.