Keywords: Coordinate-Independent Estimation, Dimension Reduction, Semiparametric Modeling
A number of dimension reduction methods for semi-parametric multi-index modeling can be formulated as a generalized eigenvalue decomposition problem. Sparse estimation the eigenvectors that constitute the basis direction of the dimension reduction space is often obtained for each of the directions independently. However, such sparse estimation approaches do not yield estimate that are invariant to orthogonal transformation of basis matrix that represents the dimension reduction subspace. We exploit a group-dantzig type formulation to obtain coordinate-independent sparse estimates that are invariant under orthogonal transformation of the dimension reduction subspace. Extensive simulation and real data application will be presented to demonstrate the effectiveness of the proposed method and compare its performance with other competing methods. Consistency of the estimates are also established.