Summary: This paper is intended as an investigation of estimating cause-specific cumulative hazard and cumulative incidence functions in a competing risks model. The proportional model in which ratios of the cause-specific hazards to the overall hazard are assumed to be constant (independent of time) is a well-known semiparametric model. We are here concerned with relaxation of the proportionality assumption. The set \(C\) of all causes is decomposed into two disjoint subsets of causes as \(C=C_{1}\cup C_{2}\). The relative risk of cause A in the sub-causes \(C_{1}\) can be represented as a function defined by the ratio of the cause-specific hazard of cause A to the sum of cause-specific hazards in the sub-causes \(C_1\). We call this function the risk pattern function of cause A in \(C_1\), and consider a semiparametric model in which risk pattern functions in \(C_1\) are not constant (independent of time) but whose functional forms, except for finite-dimensional parameters, are known.
Based on this model, semiparametric estimators are obtained, and estimated variances of them are derived by delta methods. We investigate asymptotic properties of the semiparametric estimators and compare them with nonparametric estimators. The semiparametric procedure is illustrated with a radiation-exposed mice data set, which represents lifetimes and causes of death of mice exposed to radiation in two different environments.