PRINCIPAL POINTS AND SELF-CONSISTENT POINTS OF ELLIPTIC DISTRIBUTIONS

• 成果类型: Article
• 署名作者: TARPEY, T; LI, LN; FLURY, BD
• 署名单位: National Center Atmospheric Research (NCAR) - USA; Indiana University System; Indiana University Bloomington; National Institute of Standards & Technology (NIST) - USA; Indiana University System; Indiana University Bloomington
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176324457
• 发表日期: 1995
• 页码: 103-112
• 关键词: curves

摘要

The k principal points of a p-variate random vector X are those points xi(1), ..., xi(k) is an element of R(p) which approximate the distribution of X by minimizing the expected squared distance of X from the nearest of the xi(j). Any set of k points y1, ..., y(k) partitions R(p) into ''domains of attraction'' D-1, ..., D-k according to minimal distance; following Hastie and Stuetzle we call y(1), ..., y(k) self-consistent if E[X\X is an element of D-j] = y(j) for j = 1, ..., k. Principal points are a special case of self-consistent points. In this paper we study principal points and self-consistent points of p-variate elliptical distributions. The main results are the following: (1) If k self-consistent points of X span a subspace of dimension q < p, then this subspace is also spanned by q principal components, that is, self-consistent points of elliptical distributions exist only in principal component subspaces. (2) The subspace spanned by It principal points of X is identical with the subspace spanned by the principal components associated with the largest roots. This proves a conjecture of Flury. We also discuss implications of our results for the computation and estimation of principal points.