Determining the dimension in sliced inverse regression and related methods

Sliced inverse regression (SIR) and principal Hessian directions aim to reduce the dimensionality of regression problems. An important step in the method is the determination of a suitable dimension. Although statistical tests based on the nullity eigenvalues are usually suggested, this article focuses on the quality of the estimation of the effective dimension reduction (EDR) spaces. Essentially, the goal is to retain only sufficiently stable subspaces, The goodness of the estimation is measured by the squared trace correlation between the subspaces of the EDR space and their estimates. Asymptotic expansions are derived and estimates deduced. Simulations give an insight on the behavior of the criterion and indicate how it can be used in practice.