CAREER: Nonparametric methods in multiple dimensions: shape restrictions, bootstrap and beyond

本文讨论了用非参数方法进行估计和推断的一些问题。近年来，非参数方法在统计学理论和实践中变得越来越流行，主要是因为它们比参数方法具有基本的优势：更大的灵活性和更“数据驱动”的特点。在这个方案中，研究者研究了这一领域统计研究的三个核心方向：(A)形状约束下的非参数函数估计，(B)利用半/非参数技术进行降维，(C)非标准问题中基于Bootstrap的推理。这项研究的主要动机是开发完全自动化的非参数程序(不需要调整参数，例如平滑带宽)，但仍然足够灵活，可以合并数据驱动的特征。这项建议的主要部分涉及适用于多变量数据的非参数方法，这一领域受到的关注相对较少，尽管经常被认为是执行真实数据分析所必需的。随着现代计算设施的进步和大规模数据集的爆炸式增长，涉及数值优化技术的更复杂/复杂的统计程序的使用正变得越来越流行。然而，对大多数这些程序的完整的理论分析在很大程度上仍然是可用的。这项研究的目的是了解这些统计过程中的一些理论和计算方面，并量化这些随机优化问题中涉及的不确定性。拟议研究的预期应用范围很广，从探测全球变暖的到来，到估计银河系中恒星的径向速度分布，再到开发二元选择模型的推断技术(计量经济学家特别感兴趣)，并将涉及与统计学家、生物统计学家、流行病学家、计量经济学家和天文学家在不同层面上的合作。

This proposal deals with some problems on estimation and inference using nonparametric methods. Nonparametric procedures have become increasingly popular in the theory and practice of statistics in recent times primarily because of their fundamental advantages over parametric methods: greater flexibility and more 'data-driven' features. In this proposal, the investigator studies three core directions of statistical research in this area: (A) Nonparametric function estimation under shape restrictions, (B) Dimension reduction using semi/non-parametric techniques, and (C) Bootstrap based inference in non-standard problems. The main motivation for this research is in developing nonparametric procedures that are completely automated (free from tuning parameters, e.g., smoothing bandwidths) but still flexible enough to incorporate data-driven features. A major part of this proposal deals with nonparametric methods applicable to multivariate data, an area that has received relatively less attention, though often felt to be necessary in performing real data analysis.With the advancement in modern computing facilities and the explosion in collection of large scale data sets, the use of more complicated/intricate statistical procedures involving numerical optimization techniques are becoming increasingly popular. However, a complete theoretical analysis of most of these procedures is still largely unavailable. This research aims at understanding the theoretical and computational aspects of some of these statistical procedures, and quantifying the uncertainties involved in such stochastic optimization problems. The intended applications of the proposed research are diverse, ranging from detecting the advent of global warming, to estimating the radial velocity distribution of stars in a galaxy, to developing inferential techniques for binary choice models (of special interest to econometricians), and would involve collaborations at different levels with statisticians, biostatisticians, epidemiologists, econometricians and astronomers.