Poisson approximations for time-changed point processes

By means of a distributional limit theorem Arjas and Haara (1987) have shown that the total hazards at a vector of failure times approximate independent exponential distributions provided the conditional probabilities of failures at fixed times are uniformly small and there is small chance of failures occuring together. This result provides a means of assessing goodness-of-fit for parametric survival models. In this paper, we provide explicit bounds on the difference of the joint distribution functions of the total hazards and those of the exponential distribution. These bounds give a convergence result analogous to Arjas and Haara (1987), but with weaker conditions. The bounds are obtained from extensions of the bounds for departure from Poissonity given in Brown (1983) and use compensator based time-transforms. In deriving the bounds, we reveal the connection between the point process defined by the compensator evaluated at the original points and time transforms of the stochastic process of counts: this connection is not direct in the case of a compensator with jumps, unlike the continuous case.