CAREER: Nonparametric methods in multiple dimensions: shape restrictions, bootstrap and beyond

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This proposal deals with some problems on estimation and inference using nonparametric methods. Nonparametric procedures have become increasingly popular in the theory and practice of statistics in recent times primarily because of their fundamental advantages over parametric methods: greater flexibility and more 'data-driven' features. In this proposal, the investigator studies three core directions of statistical research in this area: (A) Nonparametric function estimation under shape restrictions, (B) Dimension reduction using semi/non-parametric techniques, and (C) Bootstrap based inference in non-standard problems. The main motivation for this research is in developing nonparametric procedures that are completely automated (free from tuning parameters, e.g., smoothing bandwidths) but still flexible enough to incorporate data-driven features. A major part of this proposal deals with nonparametric methods applicable to multivariate data, an area that has received relatively less attention, though often felt to be necessary in performing real data analysis.With the advancement in modern computing facilities and the explosion in collection of large scale data sets, the use of more complicated/intricate statistical procedures involving numerical optimization techniques are becoming increasingly popular. However, a complete theoretical analysis of most of these procedures is still largely unavailable. This research aims at understanding the theoretical and computational aspects of some of these statistical procedures, and quantifying the uncertainties involved in such stochastic optimization problems. The intended applications of the proposed research are diverse, ranging from detecting the advent of global warming, to estimating the radial velocity distribution of stars in a galaxy, to developing inferential techniques for binary choice models (of special interest to econometricians), and would involve collaborations at different levels with statisticians, biostatisticians, epidemiologists, econometricians and astronomers.