Shape-constrained estimation and inference: robustness and adaptability

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In today's world, statistical analysis is driving exploration and discovery in science and engineering in unprecedented ways. Statistical methods with advantageous characteristics are crucial to furthering these advances.  I will conduct methodological, theoretical, and computational investigations focusing in the next five years on shape-constrained nonparametric methods. In recent years, there has been a dramatic and exciting increase in research activity in the area of shape-constraints, and as a result, shape-constrained methods have emerged as an appealing and superior alternative to traditional nonparametric smoothing methods.  By capitalizing on inherent features of shape-constrained nonparametric techniques, the new data analysis methods produced through the proposed research program will have a unique combination of performance characteristics: they will be robust, minimizing the need of the user to make model assumptions, they will be adaptive, naturally reaching similar efficiency to parametric approaches in certain user-specified cases, and they will be fully automatic, leading to naturally optimal and computationally efficient approaches. Specific technical objectives of the proposal are based on current open problems in the field of shape-constrained nonparametrics, while at the same time addressing important data analysis needs including (but not limited to) multivariate and high-dimensional data sets arising in insurance or various types of medical studies. Projects are driven by real data, incorporating observed data characteristics and collection methods for important and previously unconsidered scenarios.  The resulting methods will therefore improve the practitioner's ability to analyze, model, and extract information from data, paving the way for further scientific discovery.  The associated training program will produce highly qualified personnel (HQP) with advanced expertise in specialized modern computational, statistical, and mathematical methods.   The skills gained will enable these HQP to pursue successful academic, research, or industrial careers in fields such as statistics, finance, public health, and data science.