  The cause of failure in cohort studies that involve competing risks is
frequently incompletely observed. To address this, several methods have been
proposed for the semiparametric proportional cause-specific hazards model under
a missing at random assumption. However, these proposals provide inference for
the regression coefficients only, and do not consider the infinite dimensional
parameters, such as the covariate-specific cumulative incidence function.
Nevertheless, the latter quantity is essential for risk prediction in modern
medicine. In this paper we propose a unified framework for inference about both
the regression coefficients of the proportional cause-specific hazards model
and the covariate-specific cumulative incidence functions under missing at
random cause of failure. Our approach is based on a novel computationally
efficient maximum pseudo-partial-likelihood estimation method for the
semiparametric proportional cause-specific hazards model. Using modern
empirical process theory we derive the asymptotic properties of the proposed
estimators for the regression coefficients and the covariate-specific
cumulative incidence functions, and provide methodology for constructing
simultaneous confidence bands for the latter. Simulation studies show that our
estimators perform well even in the presence of a large fraction of missing
cause of failures, and that the regression coefficient estimator can be
substantially more efficient compared to the previously proposed augmented
inverse probability weighting estimator. The method is applied using data from
an HIV cohort study and a bladder cancer clinical trial.
