Sufficient Reductions in Regressions With Elliptically Contoured Inverse Predictors

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 &7C Y , are normal. By construction, these methods extract information from the first two conditional moments of X &7C Y ; they can only estimate linear reductions and thus form the linear sufficient dimension reduction (SDR) methodology. When var( X &7CY) is constant, E( X &7CY) contains the reduction and it can be estimated using PFC. When var( X &7CY) 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 &7C Y is elliptically contoured with parameters 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.