Resumen de Sufficient reductions in regressions with elliptically contoured inverse predictors

Efstathia Bura, Liliana Forzani

• 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|Y, are normal. By construction, these methods extract information from the first two conditional moments of X|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|Y) contains the reduction and it can be estimated using PFC. When var(X|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|Y is elliptically contoured with parameters and density gY, there is no linear nontrivial sufficient reduction except if gY 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.