  We develop an efficient estimation procedure for identifying and estimating
the central subspace. Using a new way of parameterization, we convert the
problem of identifying the central subspace to the problem of estimating a
finite dimensional parameter in a semiparametric model. This conversion allows
us to derive an efficient estimator which reaches the optimal semiparametric
efficiency bound. The resulting efficient estimator can exhaustively estimate
the central subspace without imposing any distributional assumptions. Our
proposed efficient estimation also provides a possibility for making inference
of parameters that uniquely identify the central subspace. We conduct
simulation studies and a real data analysis to demonstrate the finite sample
performance in comparison with several existing methods.
