CAREER: Nonparametric function estimation: shape constraints, adaptation, inference and beyond

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Nonparametric statistics is an area of statistics and machine learning that allows one to model and analyze datasets without making strong prior assumptions about the data. Data problems where the techniques of nonparametric statistics are useful come from a wide variety of applied areas including biology, medicine, astronomy, engineering, economics and operations research. In modern complex and large datasets, these methods are especially crucial as they enable the detection of important trends and patterns in the data that may be missed by traditional parametric statistical techniques. However there exist many unresolved issues concerning the theory, methodology and application of nonparametric methods in modern data problems. A systematic study of these issues will be undertaken in this project which will result in (a) an improved understanding (in terms of accuracy and uncertainty quantification) of many existing methods, and (b) novel methods and computational algorithms that will be useful to applied practitioners in the scientific areas mentioned above. Most of the proposed projects are collaborative and involve researchers from a diverse set of universities. The project also contains a well-developed plan of educational activities which will have a major impact on the education and training of undergraduate and graduate students at UC Berkeley in statistical research. In particular, many of the educational activities of the project are aimed towards undergraduate students, a group that is often given less importance at large research universities.Concretely, a wide range of nonparametric models will be studied, covering both regression and density estimation. In situations where empirically attractive estimators exist, an elaborate theoretical study is proposed focusing on their adaptive risk properties. In other situations, estimators and efficient computational algorithms are proposed together with an analysis of their accuracy. Important practical problems of inference and uncertainty quantification are also addressed. The specific regression problems that are investigated in this project include (a) multivariate convex regression, univariate trend filtering and additive shape constrained regression where adaptive risk properties of the natural estimators will be established, (b) multivariate trend filtering and quasi-convex regression where new estimators are provided along with efficient computational algorithms, and (c) global and pointwise inference in shape constrained estimation where uncertainty quantification will be addressed. In density estimation, the problems investigated include: (a) log-concave and mixture density estimation where maximum likelihood estimators will be studied, (b) distributionally robust optimization and nongaussian component analysis where novel methodology will be proposed based on shape-constrained density estimation, and (c) robust approaches to shape-constrained inference where new procedures will be developed.