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

Suppose all events occurring in an unknown number ( ν ) of iid renewal processes, with a common renewal distribution F, are observed for a fixed time τ , where both ν 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 ν and investigate its properties. Our results show that the proposed estimator performs well in terms of bias and asymptotic normality, while the MLE of ν derived assuming that the common renewal distribution is exponential may be seriously biased if that assumption does not hold.