Improved estimation and inference for regression models with functional components

具有未知功能成分的回归模型，如广义加性模型（见，例如，Hastie和Tibshirani， 1986, DOI: 10.1214/ss/1177013604和Wood， 2017, DOI: 10.1201/9781315370279）和功能回归模型（见，例如Ramsay和Silverman， 2005, DOI: 10.1007/b98888）在统计实践的不同领域发现了许多应用。这个博士项目将开发一些方法，旨在减少由于有限的样本量和需要引入粗糙度惩罚来控制函数估计器的平滑性而导致的功能组件估计器的偏差。我们还将考虑开发和评估关于功能组件的推断方法。我们将从估计广义加性模型的方法开始，最初尝试将Kosmidis和Lunardon （2021, arXiv:2001.03786）的减偏m估计框架扩展到在估计目标中添加自适应平滑惩罚的情况。该扩展框架或任何替代方案，我们产生减少偏差和提高推断在广义加性模型，然后将适应和应用于功能回归设置。该项目将着眼于开发新的方法，通过理论评估和全面的模拟研究，并将方法应用于现实世界的数据分析设置。

Regression models with unknown functional components, like generalized additive models (see, e.g., Hastie and Tibshirani, 1986, DOI: 10.1214/ss/1177013604 and Wood, 2017, DOI: 10.1201/9781315370279) and functional regression models (see, e.g. Ramsay and Silverman, 2005, DOI: 10.1007/b98888) are finding numerous applications in diverse areas of statistical practice. This PhD project will develop methods that aim to reduce the bias of estimators of functional components present due to finite sample sizes and the need to introduce a roughness penalty for controlling the smoothness of the functional estimators. We will also consider developing and assessing methods for inference about the functional components.We will begin with methods for estimating generalized additive models, initially attempting to extend the reduced-bias M-estimation framework of Kosmidis and Lunardon (2021, arXiv:2001.03786) to cases where an adaptive, smoothness penalty is added to the estimation objectives. That extended framework or any alternative we produce to reduce bias and improve inference in generalized additive models will then be adapted and applied to functional regression settings.The project will look at developing new methods, assessed theoretically and through comprehensive simulation studies, and applying the methodology to real-world data-analytic settings.