Constrains and Flexibility in Modeling

本项目主要研究与回归曲线估计有关的一些重要的统计学问题，这些回归曲线是已知或需要满足某些约束条件的，如单调性、凸性、周期性或范围界限。这些限制出现在各种社会、经济、生物和行为应用中。尽管有大量关于函数估计的文献，但很少有算法可用于合并约束。本项目从一种简单有效的基于B-Spline空间L1优化的方法开始。该方法的渐近性质、建模灵活性和计算复杂性与无约束问题基本相同。研究人员将扩展该方法的工作，包括自适应地选择样条空间的维度，约束的假设检验，二元响应、半参数模型和多元函数的处理，以及适当排序的百分位曲线的计算。除了方法学研究外，还将通过合作开发一个用户友好的约束B样条光顺软件。

This project focuses on some important statistical issues relating to the problem of estimating regressions curves that are known or required to satisfy certain constraints such as monotonicity, convexity, periodicity, or range bounds. These constraints arise in a variety of social, economic, biometric, and behavioral applications. Despite a vast amount of literature on function estimation, few algorithms are available for incorporating constraints. This project starts with a simple and effective method based on L1 optimization in the space of B-splines. The asymptotic properties, modeling flexibility, and computational complexity of the method are about the same as those of the unconstrained problems. The investigator will extend work on the method to include adaptive choice of dimension for the spline space, hypothesis testing on constraints, handling of binary response, semiparametric models and multivariate functions, and computation of properly ordered percentile curves. In addition to the methodological research, a user-friendly software for constrained B-spline smoothing will be developed through collaborations.