  We propose a class of dimension reduction methods for right censored survival
data using a counting process representation of the failure process.
Semiparametric estimating equations are constructed to estimate the dimension
reduction subspace for the failure time model. The proposed method addresses
two fundamental limitations of existing approaches. First, using the counting
process formulation, it does not require any estimation of the censoring
distribution to compensate the bias in estimating the dimension reduction
subspace. Second, the nonparametric part in the estimating equations is
adaptive to the structural dimension, hence the approach circumvents the curse
of dimensionality. Asymptotic normality is established for the obtained
estimators. We further propose a computationally efficient approach that
simplifies the estimation equation formulations and requires only a singular
value decomposition to estimate the dimension reduction subspace. Numerical
studies suggest that our new approaches exhibit significantly improved
performance for estimating the true dimension reduction subspace. We further
conduct a real data analysis on a skin cutaneous melanoma dataset from The
Cancer Genome Atlas. The proposed method is implemented in the R package
"orthoDr".
