Estimation of Functionals of High-Dimensional Parameters of Statisical Models

高维参数的低维特征估计是当代复杂、高维数据统计分析中的一个重要课题。虽然信息论的局限性往往使整个未知参数的可靠估计变得不可能，因为它的高维，低维特征的估计可以以更快的错误率有效地完成，这在经典统计学中很常见。这类问题经常出现在应用中，特别是当未知参数是一个大的矩阵，如量子系统的密度矩阵，而感兴趣的特征是这种矩阵的各种光谱特性时。尽管这些问题已经被研究了很多年，但对于代表感兴趣的特征的泛函的统计估计，几乎没有通用的方法。这个项目的主要目标是在一个一般的数学框架中研究函数估计问题，并发展一般的估计方法和一个全面的理论来说明函数估计中的错误率如何依赖于目标函数的基本属性，如其光滑性。该项目为在高层面统计领域培训研究生提供了新的机会，特别是通过发展研究生水平的课程和研讨会。该项目的主要重点是发展一种估计统计模型中未知高维参数的光滑泛函的高阶偏差减少方法(自助链偏差减少)。它是基于迭代Bootstrap的，可以看作是高维参数空间上某些积分方程组的一种近似求解方法。在高维高斯模型的情况下，该方法可以得到具有最优错误率的光滑泛函的估计器。本研究项目将研究各种重要的高维统计模型的这种估计量的性质，包括对数凹模型、流形上的模型、稀疏模型和量子统计中的密度矩阵估计模型。目标是确定函数估计中的最小极大最优误差率，并研究依赖于问题的泛函参数和复杂性参数的光滑度在快参数和慢非参数速率之间的相变。这需要解决一些具有挑战性的分析和概率问题，包括研究独立随机过程(随机同伦)的叠加逼近自举马尔可夫链，发展高维正态逼近和耦合方法，以及统计估计的集中界。该项目将导致对高维函数估计问题的更深入的理解，并在高维统计推理中开发各种新的概率工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Estimation of low-dimensional features of high-dimensional parameters is an important subject in contemporary statistical analysis of complex, high-dimensional data. While information-theoretic limitations often make impossible the reliable estimation of the whole unknown parameter due to its high dimensionality, estimation of low-dimensional features could be done efficiently with much faster error rates, common in classical statistics. Such problems often occur in applications, in particular, when the unknown parameter is a large matrix such as the density matrix of a quantum system, and the features of interest are various spectral characteristics of such matrices. Despite the fact that these problems have been studied for many years, there are few general approaches to statistical estimation of functionals representing the features of interest. The main goal of this project is to study functional estimation problem in a general mathematical framework and to develop general estimation methods as well as a comprehensive theory showing how the error rates in functional estimation depend on the underlying properties of the target functional such as its smoothness. The project provides new opportunities for training graduate students in the areas of high-dimensional statistics, in particular, by developing graduate level courses and seminars. The main focus of the project is on the development of a higher order bias reduction method (bootstrap chain bias reduction) in estimation of smooth functionals of unknown high-dimensional parameter of statistical model. It is based on iterative bootstrap and it could be viewed as a method of approximate solution of certain integral equations on high-dimensional parameter spaces. In the case of high-dimensional Gaussian models, this method yields estimators of smooth functionals with optimal error rates. This research project will study the properties of such estimators for a variety of important high-dimensional statistical models, including log-concave models, models on manifolds, sparse models and density matrix estimation models in quantum statistics. The goal is to determine minimax optimal error rates in functional estimation and to study the phase transition between fast parametric and slow nonparametric rates depending on the degree of smoothness of the functional and complexity parameters of the problem. This requires solving a number of challenging analytic and probabilistic problems, including the study of approximation of bootstrap Markov chains by superpositions of independent stochastic processes (random homotopies), the development of high-dimensional normal approximation and coupling methods as well as of concentration bounds for statistical estimators. The project will result in much deeper understanding of functional estimation problems in high dimensions and in the development of a variety of new probabilistic tools in high-dimensional statistical inference.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.