多个样本同时非参数近似的成本量化
Quantifying the cost of simultaneous non-parametric approximation of several samples
摘要：
我们考虑具有高斯误差但数据由不同样本组成的标准非参数回归模型。要回答的问题是样本是否可以由相同的回归函数充分表示。为此，我们为每个样本定义了一个通用的，诚实的和非渐近的回归函数的置信区域。当且仅当相应的置信区域的交集非空时，样本的任何子集都可以由相同的函数表示。如果样本的经验支持是不相交的，那么置信区域的交集总是非空的，并且只能通过在联合近似上设置形状或定量平滑条件来获得否定的答案。可替代地，可以计算最简单的联合近似函数，其给出联合近似的成本的度量，例如，所需的额外峰值的数量。
英文摘要：
We consider the standard non-parametric regression model with Gaussian errors but where the data consist of different samples. The question to be answered is whether the samples can be adequately represented by the same regression function. To do this we define for each sample a universal, honest and non-asymptotic confidence region for the regression function. Any subset of the samples can be represented by the same function if and only if the intersection of the corresponding confidence regions is non-empty. If the empirical supports of the samples are disjoint then the intersection of the confidence regions is always non--empty and a negative answer can only be obtained by placing shape or quantitative smoothness conditions on the joint approximation. Alternatively a simplest joint approximation function can be calculated which gives a measure of the cost of the joint approximation, for example, the number of extra peaks required.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
