Efficient and robust inference for regularization with regular and functional data

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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.