Abstract
In this study, we consider a process where states 0 and 1 appear alternately. Rootzén & Zholud (2016) developed methods to estimate the parameters of the distribution of length of 0-interval in the alternating renewal process. Abe & Kamakura (2016) derived the likelihood function corresponding to window censored alternating renewal process. Such a 0-1 process has been observed in the observation window. The specific events can only be observed in this window. When the event does not occur until the window end, the right censoring can be observed. The left censored observation is defined when the event occurs at the left end of the window.
Rootzén & Zholud (2016) only focused on the distribution of length of state 0 and discussed this problem based on the conditional likelihood. Abe & Kamakura (2016) proposed joint estimators of parameters of the distribution of length of state 1 and 0 based on the maximum likelihood. For a short observation window, Rootzén & Zholud (2016) suggested that the joint parameter estimation of the distribution of length of states 0 and 1 is preferable. However, there are no theoretical considerations or numerical comparisons performed in this regard. In this article, we investigate the properties of theses parameters, which specify the distribution of the two-state length. When the length has an exponential distribution, we can derive the asymptotic relative efficiency of our estimator to the estimator proposed by Rootzén & Zholud (2016). Even though the distribution of lengths of states 0 and 1 is independent, the joint estimation approach yields better performance. In the case of Weibull and exponential distributions we illustrate usefulness of the unconditional method (Abe & Kamakura, 2016) via simulations. As an example of application, we analyze the heat seal data observed by a microscope.
