An empirical process view of inverse regression

Most of the methods among the inverse regression literature rely on a slicing of the range of the response variable. Theoretical results are usually shown assuming that (i) the slices are fixed while in practice estimators are constructed with (ii) random slices that contain the same number of observations. In this paper we obtain the asymptotic normality in the case where the slices contains the same number of observations. This issue matter since we find a gap between the asymptotic distributions related to both approaches (i) and (ii). Along this line, we revisit the asymptotic properties of existing methods such as sliced inverse regression and cumulative inverse regression, and we also introduce a bootstrap procedure that reproduce accurately the law of certain Cramér-von Mises test statistics. Our approach is based on the stochastic analysis of some empirical processes that lie close to a certain subspace of interest called the central subspace.