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标模型，都是带有协变量的模型。研究中通常包含许多协变量，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响应变量真正相关。例如，在医学实验中，通常采用将协变量与治疗效果联系起来的特定模型，更多的是为了解释的方便和简单，而不是为了有效性。正则化方法对于识别与响应相关联的变量子集和同时进行参数估计是有用的。有效的变量选择也可以导致具有更好的预测准确性和更容易解释的简约模型。近年来，大量的研究致力于这一领域，许多鲁棒程序也被研究。(Here词语“鲁棒的”指的是过程在模型错误指定下和/或当存在异常值时保持其有效性的能力。这些方法在处理污染数据方面取得了不同程度的成功。在统计推断中需要良好的鲁棒程序现在已经被广泛认识到。实际应用中的一个常见问题是数据中存在离群值。此外，统计模型只是对现实的近似，真实的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有效和鲁棒的正则化方法。

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.