Estimation of Convex Objects

在这个建议中，凸性和统计之间的相互作用的一个方面是解决非参数统计估计问题的研究，其中凸性主要是作为一个约束控制未知的对象的兴趣。注意力集中在四个突出的问题，包括凸回归和对数凹密度估计。这些问题的重要性在统计界以及其他各学科中得到了充分的承认，并就此撰写了许多论文。然而，这些问题的估计理论和方法还存在许多尚未解决的问题，尤其是在多维情况下。事实上，这里的理论和方法论并不像非参数统计的某些其他领域那样复杂，例如平滑和稀疏约束下的经典函数估计。本提案的主要目标是弥合这一差距。重点是以下研究领域：（a）研究常用估计量如极大似然估计和最小二乘估计的理论性质;（B）建立极大极小理论（确定极小极大收敛速度，构造近似极小极大估计量等），（c）理解自适应估计，（d）实现用于计算极小极大和自适应估计量的实用算法，以及（e）基于非参数函数估计的经典思想（例如平滑和基于核的估计）构造替代的更简单的估计。我们的分析方法涉及凸几何，经验过程，非参数统计和信息论的思想。 近年来，从凸几何和最优化领域，统计学的思想和方法大量涌入。一个主要的原因是在当代应用统计学的大数据集的优势，有效的计算是必要的，凸优化技术是为这些应用量身定制的。这个建议的目的是进一步了解凸性和统计之间的这种深刻的联系，通过集中在统计问题，凸性是目前作为一个约束控制未知对象的兴趣。本研究的主要目标是使这一重要统计领域的理论和方法发展达到与其他相关统计领域（如经典函数估计）相同的复杂程度。拟议的研究在从工程到经济的各种领域都有应用。具体而言，这里研究的问题出现在诸如计算机断层扫描，从激光雷达测量的目标重建，机器人触觉传感，图像分析，几何断层扫描，经济学和运筹学中的生产估计，效用和需求/供给函数，分散检测等领域。凸几何和理论统计。

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.