Invariance properties of Sliced Inverse Regression

In this paper we consider a semiparametric regression model involving a $p$-dimensional explanatory variable ${\mathbf{x}}$ and including a dimension reduction of ${\mathbf{x}}$ via an index $B'{\mathbf{x}}$. In this model, the main goal is to estimate $B$ and to predict the real response variable $Y$ conditionally to ${\mathbf{x}}$. A standard approach is based on sliced inverse regression (SIR). We exhibit some invariance properties of this method.