Sliced Inverse Regression for 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'b. We propose an approach based on sliced inverse regression in order to estimate the space spanned by the common dimension reduction direction b. We establish root square 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 b is proposed