  We consider a class of Cox models with time-dependent effects that may be
zero over certain unknown time regions or, in short, sparse time-varying
effects. The model is particularly useful for biomedical studies as it
conveniently depicts the gradual evolution of effects of risk factors on
survival. Statistically, estimating and drawing inference on infinite
dimensional functional parameters with sparsity (e.g., time-varying effects
with zero-effect time intervals) present enormous challenges. To address them,
we propose a new soft-thresholding operator for modeling sparse, piecewise
smooth and continuous time-varying coefficients in a Cox time-varying effects
model. Unlike the common regularized methods, our approach enables one to
estimate non-zero time-varying effects and detect zero regions simultaneously,
and construct a new type of sparse confidence intervals that accommodate zero
regions. This leads to a more interpretable model with a straightforward
inference procedure. We develop an efficient algorithm for inference in the
target functional space, show that the proposed method enjoys desired
theoretical properties, and present its finite sample performance by way of
simulations. We apply the proposed method to analyze the data of the Boston
Lung Cancer Survivor Cohort, an epidemiological cohort study investigating the
impacts of risk factors on lung cancer survival, and obtain clinically useful
results.
