Shrinkage Estimation in Modern Statistics

技术描述：当代统计学处理日益庞大和复杂的数据集合。相辅相成的风格正在涌现，以适当地解释这些数据。一种通用风格涉及为数据构建多方面模型。这些应由相对较简单的组件构建，以处理情况的不同方面，但在结构的较高级别具有额外的建模参数，以便在组件之间提供连接，并提供灵活性和稳健性措施，以缓冲模型可能过于严格或部分不合适的可能性。根据上下文，这些附加的更高阶量可以用数字、定性或未知函数来描述。它们有不同的名称，如超参数或潜在变量，但无论术语及其主观解释如何，它们在数据分析方面发挥着类似的作用。将这样的高阶量添加到成分模型中，可以在它们分析缩水估计和其他形式的全球平均或模式推断时产生效果。该提案的主旨是从更根本的、组成部分的角度来理解收缩现象。查尔斯·斯坦在独立正态均值估计中发现了收缩的优势，这是上个世纪最令人惊讶和最重要的统计发展之一。正如该提案强调的那样，这一发现可以在由更高水平的量连接在一起的独立部分的框架内准确地解释。在过去的50年里，已经发展了大量的理论，以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，例如那些被标记为随机效应模型的技术，或者分层或经验的客观贝叶斯分析。然而，这一理论未能充分解决在其能够充分和适当地应用于现代复杂环境之前需要理解的一些问题。例如，经典理论倾向于将参数组分成不同的块，最好的情况是在每个块内分别收缩。但这将表明，跨区块的额外收缩是如何有益的，并建议在这方面进行进一步的研究。这项建议试图纠正的另一个经典缺陷是，当前的收缩理论相对不完整，在涉及不等抽样方差(异方差)的不平衡数据情况下有点不令人满意，但这些情况在所有高度复杂的现代数据中都是典型的。一般描述：现代统计应用程序通常涉及大量数据。对如此庞大的数据集进行概念性组织和解释是一个根本的挑战。它通常涉及对数据进行建模，使其在概率上依赖于控制被调查过程的参数。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征被很好地理解和控制的参数集合作为原始参数。这一建议引入了集合风险，以更好地解决这一问题，并建议根据估计者相对于这种集合风险的表现来判断估计者。本提案中将要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，但可以确定主要应用领域是物理和生物科学领域，如基因组学、气候学和天文学，在这些领域中，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实例，本提案中的定向和概念化部分地源自电话呼叫中心业务和互联网业务以及入侵的先前数据建模，并且当前提案的一部分涉及在涉及预测房价的不同的、复杂的环境中建模。

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.