Sufficient Reductions in Regressions With Elliptically Contoured Inverse Predictors

• 成果类型: Article
• 署名作者: Bura, Efstathia; Forzani, Liliana
• 署名单位: George Washington University; National University of the Littoral; National University of the Littoral; Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET)
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2014.914440
• 发表日期: 2015
• 页码: 420-434
• 关键词: principal hessian directions; Dimension Reduction; least-squares; Visualization; matrix; vector; robust

摘要

There are two general approaches based on inverse regression for estimating the linear sufficient reductions for the regression of Y on X: the moment-based approach such as SIR, PIR, SAVE, and DR, and the likelihood-based approach such as principal fitted components (PFC) and likelihood acquired directions (LAD) when the inverse predictors, X vertical bar Y, are normal. By construction, these methods extract information from the first two conditional moments of X vertical bar Y; they can only estimate linear reductions and thus form the linear sufficient dimension reduction (SDR) methodology. When var(X|Y) is constant, E(X vertical bar Y) contains the reduction and it can be estimated using PFC. When var(X vertical bar Y) is nonconstant, PFC misses the information in the variance and second moment based methods (SAVE, DR, LAD) are used instead, resulting in efficiency loss in the estimation of the mean-based reduction. In this article we prove that (a) if X vertical bar Y is elliptically contoured with parameters (mu(gamma), Delta) and density g(Y), there is no linear nontrivial sufficient reduction except if g(Y) is the normal density with constant variance; (b) for nonnormal elliptically contoured data, all existing linear SDR methods only estimate part of the reduction; (c) a sufficient reduction of X for the regression of Y on X comprises of a linear and a nonlinear component.