The additive risk model provides an alternative modelling technique for failure time data to the proportional hazards model. In this article, we consider the additive risk model with a nonparametric risk effect. We study estimation of the risk function and its derivatives with a parametric and an unspecified baseline hazard function respectively. The resulting estimators are the local likelihood and the local score estimators. We establish the asymptotic normality of the estimators and show that both methods have the same formula for asymptotic bias but different formula for variance. It is found that, in some special cases, the local score estimator is of the same efficiency as the local likelihood estimator though it does not use the information about the baseline hazard function. Another advantage of the local score estimator is that it has a closed form and is easy to implement. Some simulation studies are conducted to evaluate and compare the performance of the two estimators. A numerical example is used for illustration.
Bibliographical noteFunding Information:
Liu’s research is partly supported by Hunan Provincial Natural Science Foundation of China (08jj6039). Lu’s research was partly supported by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Qi’s research was supported by NSF Grant DMS-1005345. We would like to thank the Associate Editor and the referee for their constructive comments.
- Additive risk model
- Asymptotic normality
- Counting process
- Local likelihood estimator
- Local score estimator