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

On efficient dimension reduction with respect to a statistical functional of interest