Shrinkage Estimation in Modern Statistics

现代统计学中的收缩估计

• 批准号: 0707033
• 负责人: Lawrence Brown
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707033&HistoricalAwards=false
• 关键词: Shrinkage; Estimation; Modern; Statistics

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.

技术描述：当代统计学处理越来越大和复杂的数据集合。为了恰当地解释这些数据，互补的风格正在出现。一种通用的风格涉及为数据构建多面模型。这些应该从相对简单的组件构建，以处理情况的不同方面，但在结构的更高级别上使用额外的建模参数，以便提供组件之间的连接，并提供灵活性和健壮性度量，以缓冲模型可能过度限制或部分不适当的可能性。根据具体情况，这些附加的高阶量可以用数值、定性或未知函数来描述。它们有各种各样的名称，例如超参数或潜在变量，但不管术语和它们的主观解释如何，它们相对于数据分析起着类似的作用。将这样的高阶量添加到组件模型中，会影响它们对缩小估计的分析，以及对全球平均或模式的其他形式的推断。该提案的主要目的是从更基本的、组件的角度来理解收缩现象。查尔斯·斯坦（Charles Stein）在估计独立正态均值时发现了收缩的优点，这是上个世纪最令人惊讶和最重要的统计学发展之一。正如提案所强调的那样，这一发现可以准确地解释为由更高水平的量连接在一起的独立碎片的框架。在过去的50年里，已经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吸引人的技术，如那些被标记为“随机效应模型”或分层或经验客观贝叶斯分析的技术。然而，这一理论未能充分解决一些需要理解的问题，然后才能充分和适当地应用于现代复杂的环境。例如，经典理论倾向于将参数组分解成单独的块，并且在最好的情况下，在每个块中分别收缩。但它将显示如何在块间额外收缩是有益的，并在这方面提出了进一步的研究。另一个经典缺陷，这一建议的重点是试图纠正的是，目前的收缩理论是相对不完整的，在涉及不等抽样方差（异方差）的不平衡数据情况下，有些令人不满意，但这些是典型的在所有高度复杂的现代数据。一般描述：现代统计应用通常涉及大量数据。对如此大的数据集进行概念性组织和解释是一个根本性的挑战。它通常涉及将数据建模为概率依赖于控制所调查过程的参数。经典的统计公式通常将单个参数视为基本结构量，而不是将其联合建模特性很好地理解和控制的参数集合作为基本结构量。该建议引入了集成风险来更好地解决这个问题，并建议根据相对于集成风险的性能来判断评估器。本提案中所发展的理论和方法的应用几乎涵盖了所有科学和技术领域，但主要应用可以在物理和生物科学领域，如基因组学、气候学和天文学，在这些领域，大量数据集和相关参数的集合以自然的方式出现。作为潜在应用范围的进一步实例，本提案中的方向和概念化部分源于以前对电话呼叫中心流量和互联网流量和入侵的数据建模，而当前提案的一部分涉及在涉及预测房价的不同复杂背景下进行建模。