  In this paper we offer a complete methodology for sufficient dimension
reduction called the test function (TF). TF provides a new family of methods
for the estimation of the central subspace (CS) based on the introduction of a
nonlinear transformation of the response. Theoretical background of TF is
developed under weaker conditions than the existing methods. By considering
order 1 and 2 conditional moments of the predictor given the response, we
divide TF in two classes. In each class we provide conditions that guarantee an
exhaustive estimation of the CS. Besides, the optimal members are calculated
via the minimization of the asymptotic mean squared error deriving from the
distance between the CS and its estimate. This leads us to two plug-in methods
which are evaluated with several simulations.
