Statistical evaluation of multiple process data in geometric processes with exponential failures

The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.