Bracketing numbers of convex and m -monotone functions on polytopes
Bracketing numbers of convex and m -monotone functions on polytopes
复制标题
多面体上凸函数和 m 单调函数的包围数
DOI：
10.1016/j.jat.2020.105425
复制
发表时间：
2020
影响因子：
0.9
通讯作者：
Doss, Charles R.
中科院分区：
文献类型：
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作者：
Doss, Charles R.
研究了Lp范数下有界凸函数空间的包围覆盖数.括号数是理解许多统计非参数估计量渐近行为的关键量。上确界距离的括号数上界是已知的有界类，也有一个固定的Lipschitz约束。然而，在大多数感兴趣的设置中，出现的类不包括Lipschitz约束，因此不能使用基于已知括号数的标准技术。本文给出了任意多面体上无Lipschitz约束的凸函数类的括号数的上界。我们的结果是特别感兴趣的凸形状约束的基础上，在许多多维估计问题。此外，我们展示了我们的证明方法的其他应用，特别是我们定义了一类新的多元函数，所谓的m-单调函数。这样的功能已被认为是数学和统计的单变量的情况下，但从来没有在多变量的情况下。我们如何证明凸括号上界也适用于m-单调的情况。
We study bracketing covering numbers for spaces of bounded convex functions in the L p norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supremum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes that arise do not include Lipschitz constraints, and so standard techniques based on known bracketing numbers cannot be used. In this paper, we find upper bounds for bracketing numbers of classes of convex functions without Lipschitz constraints on arbitrary polytopes. Our results are of particular interest in many multidimensional estimation problems based on convexity shape constraints. Additionally, we show other applications of our proof methods; in particular we define a new class of multivariate functions, the so-called m-monotone functions. Such functions have been considered mathematically and statistically in the univariate case but never in the multivariate case. We show how our proof for convex bracketing upper bounds also applies to the m-monotone case.