Sufficient dimension reduction based on the Hellinger integral: a general, unifying approach

Sufficient dimension reduction provides a useful tool to study the dependence between a response Y and a multidimensional regressor X , sliced regression (Wang and Xia, 2008) being reported to have a range of advantages – estimation accuracy, exhaustiveness and robustness – over many other methods. A new formulation is proposed here based on the Hellinger integral of order two – and so jointly local in (X,Y ) – together with an efficient estimation algorithm. The link between χdivergence and dimension reduction subspaces is the key to our approach, which has a number of strengths. It requires minimal (essentially, just existence) assumptions. Relative to sliced regression, it is faster, allowing larger problems to be tackled, more general, multidimensional (discrete, continuous or mixed) Y as well as X being allowed, and includes a sparse version enabling variable selection, while overall performance is broadly comparable, sometimes better (especially, when Y takes only a few discrete values). Finally, it unifies three existing methods, each being shown to be equivalent to adopting suitably weighted forms of the Hellinger integral.