Nonparametric estimation of the number of components of a superposition of renewal processes

Suppose all events occurring in an unknown number (ν)(ν) of iid renewal processes, with a common renewal distribution F  , are observed for a fixed time ττ, where both νν and F   are unknown. The individual processes are not known a priori, but for each event, the process that generated it is identified. For example, in software reliability application, the errors (or bugs) in a piece of software are not known a priori, but whenever the software fails, the error causing the failure is identified. We present a nonparametric method for estimating νν and investigate its properties. Our results show that the proposed estimator performs well in terms of bias and asymptotic normality, while the MLE of νν derived assuming that the common renewal distribution is exponential may be seriously biased if that assumption does not hold.