  In this paper, we introduce principal asymmetric least squares (PALS) as a
unified framework for linear and nonlinear sufficient dimension reduction.
Classical methods such as sliced inverse regression (Li, 1991) and principal
support vector machines (Li, Artemiou and Li, 2011) may not perform well in the
presence of heteroscedasticity, while our proposal addresses this limitation by
synthesizing different expectile levels. Through extensive numerical studies,
we demonstrate the superior performance of PALS in terms of both computation
time and estimation accuracy. For the asymptotic analysis of PALS for linear
sufficient dimension reduction, we develop new tools to compute the derivative
of an expectation of a non-Lipschitz function.
