Estimation of Convex Objects

在该方案中，讨论了凸性与统计之间相互作用的一个方面，其中研究了非参数统计估计问题，其中凸性在很大程度上是作为控制感兴趣的未知对象的约束而存在的。重点研究了凸回归和对数凹密度估计等四个突出问题。统计界以及其他各种学科都充分认识到这些问题的重要性，并撰写了许多关于这些问题的论文。然而，对于这些问题的估计理论和方法还存在许多悬而未决的问题，特别是在多维情况下。事实上，这里的理论和方法远不如非参数统计的某些其他领域的理论和方法复杂，例如在光滑性和稀疏性约束下的经典函数估计。这项提议的主要目标是弥合这一差距。(A)研究最大似然估计和最小二乘估计等常用估计量的理论性质，(B)建立极小极大理论(确定极小极大收敛速度，构造近似极小极大估计量等)，(C)了解自适应估计，(D)实现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，如光滑化和基于核的估计，构造替代的更简单的估计量。我们的分析方法涉及到凸几何、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涌入了大量的统计思想和方法。这其中的一个主要原因是大型数据集在当代应用统计学中的优势，在那里需要有效的计算，而凸优化技术是为这种应用量身定做的。这项建议旨在通过侧重于凸性作为控制未知感兴趣对象的约束的统计问题，来加深我们对凸性和统计学之间的这种深刻联系的理解。这项研究的主要目的是使这一重要统计学领域的理论和方法发展达到其他相关统计学领域的成熟程度，如经典函数估计。这项拟议的研究在从工程学到经济学的多个领域都有应用。具体地说，这里研究的问题出现在计算机层析成像、激光雷达测量的目标重建、机器人触觉传感、图像分析、几何层析、产量估计、经济学和运筹学中的效用和需求/供给函数、分散检测等领域。这些研究成果还将有助于逼近理论、凸几何和理论统计等数学领域的研究。

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.