High-dimensional statistical inference in parametric and nonparametric models

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Recent advances in technology, engineering, and computing power, as well as problems in diverse areas such as genomics, clinical trials, cosmology, and climate studies have given rise to new types of inference problems that involve a very large number of unknown parameters and are known as high-dimensional inference or sparse inference problems. In general, it is difficult to treat these problems fully nonparametrically and provide procedures with nearly exact theoretical properties. The aim of the research proposal is to develop new and improve some existing inferential procedures of high-dimensional statistics. In doing so, the emphasis is on providing optimal and adaptive (not requiring the knowledge of unknown parameters of the statistical models) procedures in such areas of mathematical statistics as estimation theory, hypothesis testing theory, variable selection and classification. The main approach to be taken is nonparametric. This approach, adopted by mathematical statisticians on a worldwide scale, assumes that the parameter(s) entering the statistical models under study are infinite-dimensional. For instance, in nonparametric regression analysis, an unknown regression function mixed with weak noise is assumed to be a member of a large (infinite-dimensional) class of functions, rather than a known function depending on a finite number of unknown parameters, as in parametric regression analysis. The main criteria of goodness of a statistical procedure employed in this study is asymptotic minimaxity. This strong notion of optimality is commonly used in modern nonparametric statistical inference. We anticipate that the methods developed during the completion of this research proposal will find their usage in diverse fields such as clinical trials, astrophysics, economics, and information technology.