  We introduce a new sufficient dimension reduction framework that targets a
statistical functional of interest, and propose an efficient estimator for the
semiparametric estimation problems of this type. The statistical functional
covers a wide range of applications, such as conditional mean, conditional
variance and conditional quantile. We derive the general forms of the efficient
score and efficient information as well as their specific forms for three
important statistical functionals: the linear functional, the composite linear
functional and the implicit functional. In conjunction with our theoretical
analysis, we also propose a class of one-step Newton-Raphson estimators and
show by simulations that they substantially outperform existing methods.
Finally, we apply the new method to construct the central mean and central
variance subspaces for a data set involving the physical measurements and age
of abalones, which exhibits a strong pattern of heteroscedasticity.
