Superefficient Fits to Linear Models

最近关于收缩估计的理论、信号处理技术和计算标准正交基的有效算法使得利用经典信息界的超效率漏洞进行估计成为可能。在线性回归中，如果回归基中的前几个向量非常接近未知的平均向量，那么与不重要的基向量相关的回归系数缩小到零的估计量的风险可能比最小二乘估计量的风险要小得多。这种收缩估计器是特殊的对称线性平滑器，实现了C. Stein和M. S. Pinsker在估计高维或无限维参数方面的开创性思想的好处。研究的具体目标是：(a)构建和解释以超高效拟合为中心的置信集；(b)通过多次收缩处理所选基稀疏但不有序的情况；(c)发展观测内和观测间的收缩技术，以处理多元线性模型；(d)利用与使用离散余弦基或小波基的信号处理技术的关系。通过最小二乘方法拟合的回归模型广泛应用于科学研究和其他学科，以建立数据集内的定量关系。与环境与全球变化项目和制造业项目相关的研究就是例子。提出的研究的广泛目标是通过用更好的自适应线性平滑器代替最小二乘法来提高这些拟合关系的可靠性。最新的统计理论支持这个项目的总体可行性。如何实现理论上的可能性是这项工作的本质。作者的REACT方法（在提案中有引用）是一种实用的技术，用于使用一个响应变量进行回归，它展示了比最小二乘拟合更实际的改进。REACT与当前的非参数回归方法竞争得很好，同时提供了某些优势，例如内置的诊断，表明拟合的质量。拟议的研究将扩展REACT方法来寻找变量集之间的关系，并将开发实用的方法来评估估计关系的不确定性。最小二乘回归是现代统计软件包和电子表格中的标准函数，广泛应用于数据分析。这种情况为改进最小二乘提供了强大的动力。

Recent theory for shrinkage estimators, techniques from signal-processing, and effective algorithms for computing orthonormal bases now make it possible to exploit the superefficiency loophole in classical information bounds for estimation. In linear regression, if the first few vectors in the regression basis closely approximate the unknown mean vector, then the risk of an estimator that shrinks to zero those regression coefficients associated with the unimportant basis vectors can be much smaller than the risk of the least squares estimator. Such shrinkage estimators, which are particular symmetric linear smoothers, realize the benefits of C. Stein's and M. S. Pinsker's pioneering ideas on estimation of high- or infinite-dimensional parameters. Specific goals of the research are: (a) to construct and interpret confidence sets centered at a superefficient fit; (b) to handle, through multiple shrinkage, cases where the chosen basis is sparse but not well-ordered; (c) to develop within- and between-observation shrinkage techniques to handle the multivariate linear model; (d) to draw on relations with signal-processing techniques that use the discrete cosine basis or wavelet bases. Regression models fitted by the method of least squares are widely used in scientific research and other disciplines to establish quantitative relationships within sets of data. Studies related to the program on Environment and Global Change and to the program on Manufacturing are examples. The broad goal of the proposed research is to improve the reliability of these fitted relationships by replacing least squares with better adaptive linear smoothers. Recent statistical theory supports the general feasibility of this project. How to realize what is possible in theory is the essence of the work. The author's REACT method, described with references in the proposal, is a practical technique for regression with one response variable that demonstrates real-worldimprovements over least squares fits. REACT competes well with current nonparametric regression methods while offering certain advantages, such as built-in diagnostics that indicate the quality of the fit. The proposed research will extend REACT methods to finding relationships among sets of variables and will develop practical methods for assessing the uncertainty of the estimated relationships. Least squares regression, a standard function in modern statistical packages and spreadsheets, is widely used in data-analysis. This circumstance provides strong motivation for improving on least squares.