A Study on Nonparametric Inference Approaches for Stochastic Point Processes and Their Reliability Applications
Yasuhiro Saito · 2016
Stochastic point processes can be used as a powerful tool to describe stochastic behaviors of cumulative number of events occurred as time goes by. The occurrence of failures in repairable systems and the detection of faults in software testing are modeled by representative stochastic point processes. Nonhomogeneous Poisson process (NHPP) is well known as the simplest but most useful method for modeling such phenomena. Stochastic point processes are characterized by a conditional intensity function or the corresponding cumulative intensity function which is called mean value function especially for an NHPP. By assuming whether we can know the intensity function (or the corresponding cumulative intensity function) or not, two types of statistical inference approaches are considered. If the intensity function is known in advance, the model with the parametric intensity function is called parametric model. On the other hand, if the intensity function is unknown completely, it is called nonparametric model. In this thesis, we mainly consider nonparametric estimation methods for stochastic point processes which include NHPP and a more generalized stochastic point process called the trend renewal process. In details, we discuss several nonparametric approaches for two different research areas; preventive maintenance scheduling problem of repairable systems and software reliability assessment. In Chapter 2 and Chapter 3, we focus on parametric and nonparametric estimation methods for a periodic replacement problem with minimal repair which is a representative preventive maintenance scheduling problem. By modeling the occurrence of failures in repairable systems with NHPPs, we obtain the optimal periodic replacement time and its corresponding long-run average cost per unit time. We also discuss not only point estimation but also interval estimation for the same problem by applying several bootstrap techniques. It is revealed which method is an appropriate one in the both viewpoints of point estimation and interval estimation, throughout our simulation experiments and real failure data analyses. In Chapter 4 and Chapter 5, we pay our attention to the software reliability assessment. Since NHPP-based Software reliability models (SRMs) are widely