A Sliced Inverse Regression Approach for a Stratified Population

In this article, we consider a semiparametric single index regression model involving a real dependent variable Y, a p-dimensional quantitative covariable X, and a categorical predictor Z which defines a stratification of the population. This model includes a dimension reduction of X via an index X'beta. We propose an approach based on sliced inverse regression in order to estimate the space spanned by the common dimension reduction direction beta. We establish root n-consistency of the proposed estimator and its asymptotic normality. Simulation study shows good numerical performance of the proposed estimator in homoscedastic and heteroscedastic cases. Extensions to multiple indices models, q-dimensional response variable, and/or SIR alpha-based methods are also discussed. The case of unbalanced subpopulations is treated. Finally, a practical method to investigate if there is or not a common direction beta is proposed.