  Moment-based sufficient dimension reduction methods such as sliced inverse
regression may not work well in the presence of heteroscedasticity. We propose
to first estimate the expectiles through kernel expectile regression, and then
carry out dimension reduction based on random projections of the regression
expectiles. Several popular inverse regression methods in the literature are
extended under this general framework. The proposed expectile-assisted methods
outperform existing moment-based dimension reduction methods in both numerical
studies and an analysis of the Big Mac data.
