  Based on the theories of sliced inverse regression (SIR) and reproducing
kernel Hilbert space (RKHS), a new approach RDSIR (RKHS-based Double SIR) to
nonlinear dimension reduction for survival data is proposed and discussed. An
isometrically isomorphism is constructed based on RKHS property, then the
nonlinear function in the RKHS can be represented by the inner product of two
elements which reside in the isomorphic feature space. Due to the censorship of
survival data, double slicing is used to estimate weight function or
conditional survival function to adjust for the censoring bias. The sufficient
dimension reduction (SDR) subspace is estimated by a generalized
eigen-decomposition problem. Our method is computationally efficient with fast
calculation speed and small computational burden. The asymptotic property and
the convergence rate of the estimator are also discussed based on the
perturbation theory. Finally, we illustrate the performance of RDSIR on
simulated and real data to confirm that RDSIR is comparable with linear SDR
method. The most important is that RDSIR can also extract nonlinearity in
survival data effectively.
