In this proposal, an aspect of the interaction between convexity and statistics is addressed where nonparametric statistical estimation problems are studied in which convexity is present largely as a constraint controlling the unknown object of interest. Attention is focused on four prominent such problems including convex regression and log-concave density estimation. The importance of these problems is well recognized in the statistical community as well as various other disciplines and many papers have been written on them. However, many unsolved questions exist in the estimation theory and methodology for these problems especially in the multidimensional case. Indeed, the theory and methodology here is nowhere as sophisticated as that of certain other areas of nonparametric statistics such as classical function estimation under smoothness and sparsity constraints. The main goal of this proposal is to bridge this gap. The emphasis is on the following areas of research: (a) Studying the theoretical properties of the commonly used estimators such as MLE and least squares estimators, (b) Establishing a minimax theory (determination of the minimax rate of convergence, constructing of approximately minimax estimators etc), (c) Understanding adaptive estimation, (d) implementing practical algorithms for computing minimax and adaptive estimators, and (e) constructing alternative simpler estimators based on classical ideas from nonparametric function estimation such as smoothing and kernel based estimation. Our methods of analysis involve ideas from convex geometry, empirical processes, nonparametric statistics and information theory.
In recent years, there has been a significant influx of ideas and methods into statistics from the fields of convex geometry and optimization. A main reason for this is the predominance of large datasets in contemporary applied statistics where efficient computation is a necessity and convex optimization techniques are tailor-made for such applications. This proposal aims to further our understanding of this deep connection between convexity and statistics by focusing on statistical problems where convexity is present as a constraint controlling the unknown objects of interest. The main goal of this research is to bring the theoretical and methodological developments in this important area of statistics to the same level of sophistication present in other well-studied related areas of statistics such as classical function estimation. The proposed research has applications in a diverse set of fields ranging from engineering to economics. Specifically, the problems studied here arise in areas such as computed tomography, target reconstruction from laser-radar measurements, robotic tactile sensing, image analysis, geometric tomography, estimation of production, utility and demand/supply functions in economics and operations research, decentralized detection etc. Results coming out of this research will also contribute to the mathematical fields of approximation theory, convex geometry and theoretical statistics.
