The additive nonparametric and semiparametric Aalen model as the rate function for a counting process

We use the additive risk model of Aalen (Aalen, 1980) as a model for the rate of a counting process.
Rather than specifying the intensity, that is the instantaneous probability of an event conditional on the entire
history of the relevant covariates and counting processes, we present a model for the rate function, i.e., the
instantaneous probability of an event conditional on only a selected set of covariates. When the rate function for
the counting process is of Aalen form we show that the usual Aalen estimator can be used and gives almost
unbiased estimates. The usual martingale based variance estimator is incorrect and an alternative estimator should
be used. We also consider the semi-parametric version of the Aalen model as a rate model (McKeague and
Sasieni, 1994) and show that the standard errors that are computed based on an assumption of intensities are
incorrect and give a different estimator. Finally, we introduce and implement a test-statistic for the hypothesis of a
time-constant effect in both the non-parametric and semi-parametric model. A small simulation study was
performed to evaluate the performance of the new estimator of the standard error.