Two CUSUM schemes for simultaneous monitoring of parameters of a shifted exponential time to events

Two-parameter (shifted) exponential distribution is widely applied in many areas such as reliability modeling and analysis where time to failure is protected by a guaranty period that induces an origin parameter in the exponential model. Despite a large volume of works on inferential aspects of two-parameter exponential distribution, only few studies are done from the perspective of process monitoring. In the modern production process, where items come with a warranty, we often encounter shifted-exponential time between events from consumers' perspective, and therefore, in this paper, we propose two CUSUM schemes for joint monitoring of the origin and scale parameters based on the Maximum Likelihood estimators. We study the in-control behavior of the proposed procedures via Markov chain approach as well as applying Monte Carlo. We provide detailed implementation strategies of the two schemes along with the follow-up procedures to identify the source of shifts when an out-of-control signal is obtained. We examine the performance properties of CUSUM schemes and find that the two proposed schemes offer performance advantages over the Shewhart-type schemes especially for monitoring small to moderate shifts. Further, we provide some guidance for choosing the appropriate schemes and study the effect of reference parameter of the CUSUM schemes. We also investigate the optimal design of reference values both in known and unknown shift cases. Finally, two examples are given to illustrate the implementation of the proposed approach.