统计深度函数及其应用

次序统计量在一维统计数据分析中起着很重要的作用．多年来,人们一直在商维数据处理和分析中寻找“次序统计量”,却没有得到很满意的结果．由于缺少自然而有效的高维数据排序方法,因而象一维“中位数”的概念很难推广到高维．统计深度函数则提供了高维数据排序的一种工具,其主要思想是提供了一种从高维数据中心(最深点)向外的排序方法．不仅如此,统计深度函数已经在探索性高维数据分析,统计判决等方面带给我们一种全新的前景,并在工业、工程、生物医学等诸多领域得到很好的应用．本文介绍了统计深度函数概念及其应用,讨论了位置深度函数的标准,介绍了几种常用的统计深度函数．给出了由深度函数特别是由投影深度函数所诱导的位置和散布阵估计,介绍了它们的诸多优良性质,如极限分布,稳健性和有效性．由于在大多数场合下,高崩溃点的估计不是较有效的估计,而由统计深度函数所诱导的估计具有多元仿射不变性,并能提供理想的稳健性与有效性之间的平衡,本文还讨论了基于深度的统计检验和置信区域,介绍了统计深度函数的其他应用,如多元回归、带有变量误差模型、质量控制等,以及实际计算问题．指出了统计深度函数领域有关进一步的工作和研究方向．

Statistical Depth Functions and Some Applications

Simple one-dimensional statistics based on ordering have played such an important role in one-dimensional data analysis that their multi-dimensional analogues have been sought for years, without completely satisfactory results. The extension to higher dimensions of these one-dimensional statistics, such as the median, is difficult because there is no natural and unambiguous method of fully ordering or ranking multi-dimensional observations. Statistical depth functions are proving to be a promising tool for ordering multi-dimensional observations. The main idea of depth functions is to provide from the "deepest" point a "center-outward" ordering of multi-dimensional observations. Multi-dimensional data ordering is not the only application of depth functions, though. Depth functions have brought us new perspectives towards multi-dimensional exploratory data analysis and inference, and have shown to have significant applications in disciplines ranging from industrial engineering to biomedical sciences. This article surveys statistical depth functions and their applications. Criteria for location depth functions are discussed and some popular notions of depth are examined with emphasis on projection depth. Location and scatter estimators induced from projection depth functions are examined with a focus on their limiting distribution, robustness and efficiency. Unlike many existing high breakdown point counterparts which are ironically not very efficient, these depth induced estimators are proven to be able to keep a desirable balance between robustness and efficiency and consequently represent favorable choices of multivariate affine equivariant estimators. Depth based testing and confidence regions are discussed. Other applications of depth functions (e.g., those in regression and errors-in-variables models and in quality control) and computing issues are addressed. Future research directions in the field are highlighted.