A Framework For Estimation of Convex Functions
A Framework For Estimation of Convex Functions
复制标题
凸函数估计的框架
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Mark G. Low
中科院分区：
文献类型：
--
作者：
T. Cai;Mark G. Low
在估计某个点的凸函数的背景下，引入了一个通用的非渐近框架，该框架评估单个函数上任何过程的性能。该框架与传统的极小极大理论有显着不同，也适用于形状约束推理中的其他问题。为每个凸函数的任何估计的均方误差提供基准，就像费舍尔信息依赖于常规参数模型中的未知参数一样。引入了局部连续模，并证明它可以解决估计单个凸函数的困难。提出了一种完全数据驱动的估计器，并且证明它在每个凸函数的理想基准的恒定因子内表现一致。因此，这样的估计器适用于每个未知函数，而不是像非参数函数估计文献中典型的那样适应函数类的集合。
A general non-asymptotic framework, which evaluates the performance of any procedure at individual functions, is introduced in the context of estimating convex functions at a point. This framework, which is significantly different from the conventional minimax theory, is also applicable to other problems in shape constrained inference. A benchmark is provided for the mean squared error of any estimate for each convex function in the same way that Fisher Information depends on the unknown parameter in a regular parametric model. A local modulus of continuity is in- troduced and is shown to capture the difficulty of estimating individual convex functions. A fully data-driven estimator is proposed and is shown to perform uni- formly within a constant factor of the ideal benchmark for every convex function. Such an estimator is thus adaptive to every unknown function instead of to a col- lection of function classes as is typical in the nonparametric function estimation literature.