Application of the Bootstrap Approach to the Choice of Dimension and the α Parameter in the SIRα Method

Abstract

To reduce the dimensionality of regression problems, sliced inverse regression approaches make it possible to determine linear combinations of a set of explanatory variables X related to the response variable Y in general semiparametric regression context. From a practical point of view, the determination of a suitable dimension (number of the linear combination of X) is important. In the literature, statistical tests based on the nullity of some eigenvalues have been proposed. Another approach is to consider the quality of the estimation of the effective dimension reduction (EDR) space. The square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. In this article, we focus on the SIRα method and propose a naïve bootstrap estimation of the square trace correlation criterion. Moreover, this criterion could also select the α parameter in the SIRα method. We indicate how it can be used in practice. A simulation study is performed to illustrate the behavior of this approach.