Optimal Curve Estimation: from Asymptotic to Small Sample Sizes

Sam Efrom movich最优曲线估计：从渐近到小样本量9971051本研究的主要焦点是探索非参数曲线估计的锐渐近理论与小样本量实际情况之间的联系。主要目标如下：(I)对于经典的直接统计设置，如密度估计或异方差回归，创建了寻找最小样本量(问题的程度)的理论和方法，其中渐近尖锐的数据驱动估计优于任何其他估计。(2)对于非参数生存分析、模糊图像或回归等间接统计问题，在预测值中存在测量误差的情况下，创建这些设置与相应的直接设置之间的等价性理论，并特别强调找出类似估计质量所需的样本量之间的关系。(3)对于在万维网等现代通信系统中经常出现的具有重尾和长相依关系的时间序列的非参数曲线估计问题，发展了数据驱动的最优非参数估计方法。统计学理论主要关注的是为大样本找到最准确的程序，而在实际有趣的情况下，样本大小可能是大的也可能是小的。此外，大样本或小样本的概念取决于潜在的背景。这项研究致力于开发通用程序，该程序对大样本和小样本都是同时最优的。开发的方法将在促使提出这一建议的下列具体实际问题上进行测试：(1)二型糖尿病患者的胰岛素分泌；(2)利用海洋磁力观测的地磁极时间尺度和海底扩散速度，以及这些因素如何影响环境和全球变化；(3)饮用水质量，包括监测新墨西哥州的砷和其他致癌物质；(4)用于急诊室筛查患者的学习机；(5)分析变化和噪声图像的工程系统。这些应用问题将与桑迪亚国家实验室、新墨西哥大学医学中心和毛伊岛高性能计算中心一起完成。

SAM EFROMOVICH Optimal curve estimation: from asymptotic to small sample sizes9971051The primary focus of this research is to explore the bridges between the sharp asymptotic theory of nonparametric curve estimation and practically important cases of small sample sizes. Main objectives are as follows: (i) For the case of classical direct statistical settings, like density estimation or heteroscedastic regression, create the theory and methods of finding minimal sample sizes (the degree of the problem) where asymptotically sharp data-driven estimates outperform any other estimates. (ii) For indirect statistical problems, such as nonparametric survival analysis, blurred images or regressionwith measurement errors in predictors, create the theory of equivalence between these settings and the corresponding direct ones with a particular emphasis on finding the relation between sample sizes needed for a similar quality of estimation. (iii) For nonparametric curve estimation problems involving time series with heavy tails and long range dependence, which frequently arise in the modern communication systems like the World Wide Web, develop optimal data-driven procedures of nonparametric estimation. Statistical theory is primarily interested in finding most accurate procedures for large samples while in practically interesting cases sample sizes may be both large and small. Moreover, the notion of a large or small sample depends on an underlying setting. The research is devoted to developing universal procedures, which are simultaneously optimal for both large and small sample sizes. The developed methods will be tested on the following specific practical problems which motivated this proposal: (1) Insulin secretion by the type II diabetes patients; (2) The geomagnetic polarity time-scale and the speed of seafloor spreading using marine magnetic observations, and how these factors affect environment and global change; (3) Drinking water quality including monitoring arsenic and other cancer-causing substances in New Mexico; (4) Learning machines for screening patients in emergency rooms; (5) Engineered systems for the analysis of transformed and noisy images. These applied problems will be done together with Sandia National Laboratories, the UNM Medical Center and Maui High Performance Computing Center.