  With the development of data collection techniques, analysis with a survival
response and high-dimensional covariates has become routine. Here we consider
an interaction model, which includes a set of low-dimensional covariates, a set
of high-dimensional covariates, and their interactions. This model has been
motivated by gene-environment (G-E) interaction analysis, where the E variables
have a low dimension, and the G variables have a high dimension. For such a
model, there has been extensive research on estimation and variable selection.
Comparatively, inference studies with a valid false discovery rate (FDR)
control have been very limited. The existing high-dimensional inference tools
cannot be directly applied to interaction models, as interactions and main
effects are not ``equal". In this article, for high-dimensional survival
analysis with interactions, we model survival using the Accelerated Failure
Time (AFT) model and adopt a ``weighted least squares + debiased Lasso''
approach for estimation and selection. A hierarchical FDR control approach is
developed for inference and respect of the ``main effects, interactions''
hierarchy. { The asymptotic distribution properties of the debiased Lasso
estimators} are rigorously established. Simulation demonstrates the
satisfactory performance of the proposed approach, and the analysis of a breast
cancer dataset further establishes its practical utility.
