Efficient and robust inference for regularization with regular and functional data

许多广为人知的参数模型，包括某些线性多元回归模型、广义线性模型和大多数单指数模型，都是带有协变量的模型。通常许多协变量被包括在研究中，但由于稀疏性，这些观察到的协变量中只有一部分被认为与反应变量真正相关。例如，在医学实验中，将协变量与治疗效果相关的特定模型往往更多地是为了解释的方便和简单而不是为了有效性而采用。正则化方法对于识别与响应相关联的变量子集和同时用于参数估计是有用的。有效的变量选择还可以带来简约的模型，具有更好的预测精度和更容易的解释。近年来，人们对这一领域进行了大量的研究，也研究了许多稳健的方法。(这里的“稳健”一词指的是程序在模型错误指定和/或存在异常值的情况下保持其有效性的能力。)这些方法在处理受污染的数据方面取得了不同程度的成功。现在，人们已经广泛认识到，在统计推断中需要良好而稳健的程序。实际应用中的一个常见问题是数据中存在异常值。此外，统计模型只是对现实的近似，真实数据永远不会完全来自指定的模型。提出的研究的一个目标是开发同时有效和健壮的正则化方法。*在许多科学研究中，越来越多地遇到函数数据。功能数据由作为曲线、曲面或其他对象在时间和空间等连续体上变化的重复测量组成。近年来，功能数据分析在许多领域得到了越来越多的关注，包括从脑成像数据到神经疾病的临床诊断和纵向数据研究。例如，人脑的扩散张量成像数据是弧长方面的函数数据。另一个例子是功能磁共振成像数据，其中使用小脑血流动力学反应函数的平均变化来预测注意力缺陷多动障碍指数。在函数数据中经常会遇到离群点，包括完整的离群点曲线(全局离群点)以及具有局部离群点特征的曲线，这些特征可以在时间域或频域中局部化(局部离群点)。稳健的方法论在这些研究中很重要，因为它们代表了应用实验的结果。这一领域拟议研究的一个目标是研究函数回归模型的有效稳健估计和正则化方法，因为有效和稳健的过程对于有效的数据分析是至关重要的。

Many widely known parametric models, including certain linear multivariate regression models, generalized linear models and most single-index models, are models with covariates. Often many covariates are included in studies, but only a part of these observed covariates is believed to be truly relevant to the response variable due to sparsity. For instance, in medical experiments particular models relating covariates to treatment effects are often adopted more for convenience and simplicity of interpretation than for validity. Regularization methods are useful for identifying a subset of variables that is associated with a response and for parameter estimation simultaneously. Effective variable selection can also lead to parsimonious models with better prediction accuracy and easier interpretation. In recent years, a considerable amount of research has been devoted to this area, and many robust procedures have also been studied. (Here the word ‘robust' refers to the ability of a procedure to retain its validity under a model misspecification and/or when outliers are present.) These methods have had varying degrees of success in dealing with contaminated data. The need for good robust procedures in statistical inference has been widely recognized now. A common problem in practical applications is the presence of outliers in the data. Furthermore, statistical models are just approximations to reality and that real data never come from the specified model exactly. A goal of the proposed research is to develop regularization methods that are simultaneously efficient and robust.***In many scientific studies functional data are increasingly encountered. Functional data are made up of repeated measurements taken as curves, surfaces or other objects varying over a continuum such as time and space. Functional data analysis has gained increasing attention during recent years in many areas, including in clinical diagnosis of neurological diseases from the brain imaging data and in longitudinal data studies. For example, the diffusion tensor imaging data of human brain are functional data in terms of arc-length. Another example is the functional magnetic resonance imaging data where the averaged changes of hemodynamic response functions for cerebellum are used to predict the attention deficit hyperactivity disorder index. Outliers are frequently encountered in functional data, including entire outlying curves (global outliers) as well as curves with local outlying features, which can be localized in either the time or frequency domain (local outliers). A robust methodology is important in these studies, as they represent outcomes of applied experiments. A goal of the proposed research in this area is to investigate efficient robust estimation and regularization methods for functional regression models, as efficient and robust procedures are vital for effective data analysis.********