  Recently the shape-restricted inference has gained popularity in statistical
and econometric literature in order to relax the linear or quadratic covariate
effect in regression analyses. The typical shape-restricted covariate effect
includes monotonic increasing, decreasing, convexity or concavity. In this
paper, we introduce the shape-restricted inference to the celebrated Cox
regression model (SR-Cox), in which the covariate response is modeled as
shape-restricted additive functions. The SR-Cox regression approximates the
shape-restricted functions using a spline basis expansion with data driven
choice of knots. The underlying minimization of negative log-likelihood
function is formulated as a convex optimization problem, which is solved with
an active-set optimization algorithm. The highlight of this algorithm is that
it eliminates the superfluous knots automatically. When covariate effects
include combinations of convex or concave terms with unknown forms and linear
terms, the most interesting finding is that SR-Cox produces accurate linear
covariate effect estimates which are comparable to the maximum partial
likelihood estimates if indeed the forms are known. We conclude that concave or
convex SR-Cox models could significantly improve nonlinear covariate response
recovery and model goodness of fit.
