Shrinkage Estimation in Modern Statistics

技术说明：当代统计学处理的是日益庞大和复杂的数据集。正在出现适当解释这些数据的补充风格。一种通用的风格涉及为数据构建多方面的模型。这些应该从相对简单的组件构建，以处理情况的各个方面，但在结构的更高层次上具有额外的建模参数，以便在组件之间提供连接，并提供灵活性和鲁棒性措施，以缓冲模型可能过度限制或部分不适当的可能性。这些额外的高阶量可以根据上下文在数值上、定性地或作为未知函数来描述。它们有各种各样的名称，如超参数或潜变量，但无论术语和主观解释如何，它们在数据分析方面都起着类似的作用。将这些高阶量添加到分量模型中，会影响其对收缩估计的分析以及对全球平均值或模式的其他形式的推断。该提案的主旨是从更基本的组件角度来理解收缩现象。查尔斯·斯坦发现了收缩在估计独立正态平均值方面的优势，这是上个世纪最令人惊讶和最重要的统计学发展之一。正如该提案所强调的那样，这一发现可以在由更高层次的量联系在一起的独立部分的框架中得到解释。在过去的50年里，人们发展了大量的理论来严格理解处理这种联系的后果，以及使用某些结构上有吸引力的技术，如随机效应模型或分层或经验客观贝叶斯分析。然而，这一理论未能充分解决一些问题，需要了解它才能充分和适当地应用于现代复杂的设置。例如，经典理论倾向于将参数组分解为单独的块，并且最好在每个块内单独收缩。但它将显示如何跨块额外的收缩可以是有益的，并在这方面提出了进一步的研究。另一个经典的缺陷，这个建议集中在试图纠正的是，目前的收缩理论是相对不完整的，有些不令人满意的不平衡的数据情况下，涉及不等的抽样方差（异方差），但这些都是典型的所有高度复杂的现代data.General Description：现代统计应用往往涉及大量的数据。对如此庞大的数据集进行概念组织和解释是一项根本性的挑战。它通常涉及将数据建模为概率依赖于控制所研究过程的参数。经典的统计公式通常将单个参数视为原始结构量，而不是将其联合建模特性被很好地理解和控制的参数的原始集合。该提案引入了整体风险，以更好地解决这个问题，并建议判断估计，根据他们的性能相对于这个整体风险。本提案中拟开发的理论和方法的应用几乎包括所有科学和技术领域，但主要应用可在基因组学、气候学和天文学等物理和生物科学领域，因为在这些领域，大量数据集和相关参数集合以自然方式出现。作为潜在应用范围的另一个例子，该提案中的方向和概念化部分来自以前的电话呼叫中心流量和互联网流量和入侵的数据建模，当前提案的一部分涉及在不同的复杂背景下建模，涉及预测房价。

Technical Description: Contemporary statistics deals with increasingly large and complex collections of data. Complementary styles are emerging for appropriately interpreting such data. One general style involves constructing multi-faceted models for the data. These should be built from relatively simpler components to treat separate aspects of the situation, but with additional modeling parameters at higher levels of the structure in order to provide connections among the components and also to provide flexibility and a measure of robustness to cushion against the possibility that the model may be overly restrictive or partially inappropriate. These additional higher order quantities may be described numerically, qualitatively, or as unknown functions depending on the context. They have various names, such as hyperparameters or latent variables, but irrespective of the terminology and of their subjective interpretation they serve comparable roles relative to the analysis of the data. Adding such higher order quantities to component models has the effect in their analysis of shrinking estimates and other forms of inference toward global averages or patterns. The major thrust of the proposal is to understand the shrinkage phenomenon from a more fundamental, componentwise perspective. Charles Stein's discovery of the advantages of shrinkage in the estimation of independent normal means is among the most surprising and important statistical developments of the preceding century. As the proposal emphasizes this discovery can be interpreted in exactly the framework of independent pieces tied together by higher level quantities. A great deal of theory has been developed over the past 50 years to rigorously understand the consequences of dealing with such a connection, and of using certain structurally appealing techniques such as those labeled as random-effects models" or hierarchical or empirical objective Bayes analyses. However this theory has failed to adequately address some issues that need to be understood before it can be adequately and properly applied in modern complex settings. For example, classical theory has tended to break groups of parameters into separate blocks and, at best, to shrink separately within each block. But it will be shown how additional shrinkage across blocks can be beneficial, and further research is proposed in this regard. Another classical deficiency that this proposal focuses on trying to correct is that current theory of shrinkage is relatively incomplete and somewhat unsatisfactory with regard to unbalanced data situations involving unequal sampling variances (heteroscedasticity), but these are typical in all highly complex modern data.General Description: Modern statistical applications often involve massive amounts of data. Conceptual organization and interpretation of such large data sets is a fundamental challenge. It often involves modeling the data as being probabilistically dependent on parameters that control the process being investigated. Classical statistical formulations typically view individual parameters as the primitive structural quantities, rather than taking as primitives ensembles of parameters whose joint modeling characteristics are well understood and controlled. This proposal introduces the ensemble-risk to better address this issue and suggests judging estimators according to their performance relative to this ensemble-risk. Applications of the theory and methodology to be developed in this proposal include nearly all areas of science and technology, but principle applications can be identified in areas of physical and biological sciences such as genomics, climatology and astronomy where large data sets and ensembles of related parameters appear in a natural fashion. As a further instance of the range of potential applications, the orientation and conceptualization in this proposal derives in part from previous data modeling of telephone call-center traffic and internet traffic and intrusions, and a portion of the current proposal involves modeling in a different, complex context involving forecasting housing prices.