Inference using Shape-Restricted Regression Splines

使用形状限制回归样条进行推理

• 批准号: 0905656
• 负责人: Mary Meyer
• 金额: $ 18万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905656&HistoricalAwards=false
• 关键词: Inference; using; Shape; Restricted; Regression

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI develops methods in function estimation and inference, using shape-restricted regression splines. The work includes three broad areas in estimation and inference. First, generalized multiple regression models is investigated, where the mean response function is assumed to be smooth and have a shape restriction such as monotone or convex. Second, a new maximum-likelihood method for smoothed unimodal density estimation is developed, that allows for heavy tails such as in the Pareto family of densities. An application is a new robust regression method that estimates the error density estimation non-parametrically, simultaneously with the regression function. Finally, the proportional hazards model is developed, where the hazard function is assumed to be smooth and have a shape restriction such as monotonicity or convexity. Many problems in data analysis involve estimation of a function. Standard methods require the specification of the function up to a few parameters, but the a priori knowledge about the function is often vague and qualitative. For example, the researcher might know that a growth curve is smooth, increasing, and concave. The expected number of nesting sites at a lake might be a decreasing function of some pollution measure. Perhaps a hazard rate function is known to be increasing and convex, as in modeling wear-out of a mechanical part, or bath-tub shaped, as in modeling organ transplant failures. Nonparametric methods in function estimation are appealing because they require minimal assumptions, but development of practical inference methods is more difficult. Many methods that assume only smoothness of the function are sensitive to user-defined choices of the smoothing parameters such as bandwidth, number of knots, or penalty parameter, and the user can not rely on inference results that change with these choices. However, when the researcher can also assume a shape such as increasing or convex, the fits to the data become more robust, for the simple reason that the ``wiggling'' associated with over-fitting is obviated. The PI develops inference methods in three important areas: first, generalized regression models, such as when the response is a count or binary. Second, a new method for robust regression is developed, where the error density is assumed to be unimodal and symmetric, to allow for either heavy-tailed or thin-tailed errors. Finally, the proportional hazards model is developed, under shape and smoothness assumptions for the hazard function. These models are often used in medical studies to compare treatments while accounting for possible mitigating factors, and in industry to model mechanical systems. All three of these research projects result in basic data-analysis tools that can be used in virtually any area of science.

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。PI发展了函数估计和推理的方法，使用形状受限的回归样条法。这项工作包括估计和推断方面的三个广泛领域。首先，研究了广义多元回归模型，其中均值响应函数是光滑的，且具有单调或凸等形状约束。其次，提出了一种新的平滑单峰密度估计的极大似然方法，该方法考虑了Pareto密度族中的重尾。一个应用是一种新的稳健回归方法，它与回归函数同时非参数地估计误差密度估计。最后，建立了比例风险模型，假设风险函数是光滑的，且具有单调性或凸性等形状约束。数据分析中的许多问题都涉及函数的估计。标准方法需要指定函数，最多只需要几个参数，但关于函数的先验知识往往是模糊的和定性的。例如，研究人员可能知道增长曲线是平滑的、递增的和凹陷的。一个湖上的筑巢地点的预期数量可能是某种污染措施的递减函数。也许已知的危险率函数是递增的和凸的，就像在模拟机械部件的磨损或浴缸形状时，就像在模拟器官移植失败时一样。函数估计中的非参数方法很有吸引力，因为它们需要最小的假设，但开发实用的推理方法更加困难。许多只假设函数光滑性的方法对用户定义的平滑参数的选择很敏感，例如带宽、节点数或惩罚参数，并且用户不能依赖随着这些选择而改变的推断结果。然而，当研究人员也可以采取诸如增加或凸起的形状时，对数据的拟合变得更加稳健，原因很简单，因为避免了与过度拟合相关的“摆动”。PI在三个重要领域发展了推理方法：第一，广义回归模型，例如当响应是计数或二进制时。其次，提出了一种稳健回归的新方法，该方法假定误差密度是单峰对称的，可以同时考虑重尾误差和细尾误差。最后，在风险函数的形状和光滑性假设下，建立了比例风险模型。这些模型经常用于医学研究，以比较治疗方法，同时考虑可能的缓解因素，并在工业中用于模拟机械系统。所有这三个研究项目都产生了基本的数据分析工具，几乎可以在任何科学领域使用。