A new approach on recursive and non-recursive SIR methods

We consider a semiparametric single index regression model involving a p-dimensional quantitative covariable x and a real dependent variable y. A dimension reduction is included in this model via an index x′β. Sliced inverse regression (SIR) is a well-known method to estimate the direction of the Euclidean parameter β which is based on a “slicing step” of y in the population and sample versions. The goal of this paper is twofold. On the one hand, we focus on a recursive version of SIR which is also suitable for multiple indices model. On the other hand, we propose a new method called SIRoneslice when the regression model is a single index model. The SIRoneslice estimator of the direction of β is based on the use of only one “optimal” slice chosen among the H slices. Then, we provide its recursive version. We give an asymptotic result for the SIRoneslice approach. Simulation study shows good numerical performances of the SIRoneslice method and clearly exhibits the main advantage of using recursive versions of the SIR and SIRoneslice methods from a computational time point of view. A real dataset is also used to illustrate the approach. Some extensions are discussed in concluding remarks. The proposed methods and criterion have been implemented in R and the corresponding codes are available from the authors.