Adaptation in log-concave density estimation
Adaptation in log-concave density estimation
复制标题
对数凹密度估计的适应
作者：
Arlene K. H. Kim;Adityanand Guntuboyina;R. Samworth
实线上密度的对数凹极大似然估计，基于一个大小为n的样本，相对于平方Hellinger距离，可获得极小极大最优收敛速度。在本文中，我们证明了它也具有吸引人的适应性，即当真密度的对数为$k$-仿射(即由$k$仿射片段组成)时，只要$k$不太大，它就能获得更快的收敛速度。我们的结果使用了两种不同的技巧：第一种方法依赖于一个新的马歇尔不等式来估计对数凹密度，并且揭示了当真密度在其支撑点上接近对数线性时，对数凹极大似然估计可以在全变差距离内获得参数收敛速度。我们的第二个方法依赖于局部括号熵方法，并允许我们证明了一个尖锐的Oracle不等式，这意味着当真密度是对数凹的且其对数接近$k$-仿射时，包括Kullback-Leibler发散在内的各种全局损失函数的收敛速度是$O\Bigl(\FRAC{k}{n}\log^{5/4}n\BiGR)$。
The log-concave maximum likelihood estimator of a density on the real line based on a sample of size $n$ is known to attain the minimax optimal rate of convergence of $O(n^{-4/5})$ with respect to, e.g., squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is $k$-affine (i.e.\ made up of $k$ affine pieces), provided $k$ is not too large. Our results use two different techniques: the first relies on a new Marshall's inequality for log-concave density estimation, and reveals that when the true density is close to log-linear on its support, the log-concave maximum likelihood estimator can achieve the parametric rate of convergence in total variation distance. Our second approach depends on local bracketing entropy methods, and allows us to prove a sharp oracle inequality, which implies in particular that the rate of convergence with respect to various global loss functions, including Kullback--Leibler divergence, is $O\bigl(\frac{k}{n}\log^{5/4} n\bigr)$ when the true density is log-concave and its logarithm is close to $k$-affine.
影响因子：
4.5
作者：
Seregin,Arseni;Wellner,JonA
通讯作者：
Wellner,JonA
影响因子：
4.5
作者：
Doss CR;Wellner JA
通讯作者：
Wellner JA
当真值是线性时的凸最小二乘估计。
影响因子：
1.1
作者：
Chen,Yining;Wellner,JonA
通讯作者：
Wellner,JonA