Determining the dimension in Sliced

Sliced inverse regression and principal Hessian directions (Li, 1991, 1992) aim to reduce the dimensionality of regression problems. An important step in the method is the determination of a suitable dimension. While statistical tests based on the nullity eigenvalues are usually suggested, we here focus on the quality of the estimation of the eeective dimension reduction (edr) spaces. Essentially, our goal is to only retain suuciently 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 behaviour of the criterion and indicate how it can be used in practice.