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。所提出的研究的一个目标是开发同时高效和健壮的正则化方法。

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.