Abstract: When researchers do not have 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 that summarizes the contribution of covariates on a right-censored survival outcome. The proposed estimator determines the degree of dimension reduction adaptively from the 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. Furthermore, 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 a data-adaptive structural dimension and bandwidth. The estimation can be implemented easily using a forward selection algorithm; this implementation is justified by the asymptotic convexity of the criterion in working dimensions. Numerical simulations and two real examples are given to illustrate the proposed method.

Key words and phrases: Central subspace, counting process, data-adaptive band-width, higher-order kernel, structural dimension.