Sufficient dimension reduction with simultaneous estimation of effective dimensions for time-to-event data.

When there is not enough scientific knowledge to assume a particular regression model, sufficient dimension reduction is a flexible yet parsimonious nonparametric framework to study how covariates are associated with an outcome. We propose a novel estimator of low-dimensional composite scores, which can summarize the contribution of covariates on a right-censored survival outcome. The proposed estimator determines the degree of dimension reduction adaptively from data; it estimates the structural dimension, the central subspace and a rate-optimal smoothing bandwidth parameter simultaneously from a single criterion. The methodology is formulated in a counting process framework. Further, the estimation is free of the inverse probability weighting employed in existing methods, which often leads to instability in small samples. We derive the large sample properties for the estimated central subspace with data-adaptive structural dimension and bandwidth. The estimation can be easily implemented by a forward selection algorithm, and this implementation is justified by asymptotic convexity of the criterion in working dimensions. Numerical simulations and two real examples are given to illustrate the proposed method.