Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP’s, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next...

We analyze the notion of ‘reliability prediction’ by studying in detail a key property that is tacitly assumed to make reliability prediction possible. The analysis leads in turn to a special type of point process for which the connection of future to past can be explicitly displayed. In this type of process, the semi-renewal process, all finitedimensional distributions are completely determine...

In the first part of this paper, we give two characterizations of the multiparameter Poisson process: one characterization of poisson process using the renewal property in the space R+, and the other one using the exponential distributions of random areas of rectangles. Finally, we generalize partly a characterization of the randommeasure associated with a renewal Cox process in R+.

These notes cover material beyond the topics covered in IEOR 3106. We refer to the books by [Asmussen (2003)] and [Ross (1996)] (a more advanced book than the one we use). First, renewal theory is about renewal processes. A key quantity is the renewal function. The renewal function is important because it is a key component of the solution of the renewal equation. In applications, we are often ...

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time i...

Recent work on the statistical modeling of neural responses has focused on modulated renewal processes in which the spike rate is a function of the stimulus and recent spiking history. Typically, these models incorporate spike-history dependencies via either: (A) a conditionally-Poisson process with rate dependent on a linear projection of the spike train history (e.g., generalized linear model...

The trend-renewal-process (TRP) is defined to be a time-transformed renewal process, where the time transformation is given by a trend function λ(·) which is similar to the intensity of a nonhomogeneous Poisson process (NHPP). A nonparametric maximum likelihood estimator of the trend function of a TRP can be obtained much in the same manner as for the NHPP using kernel smoothing. But for a TRP ...

For applying life-cycle costing, the total costs should be computed over a bounded or unbounded time horizon. In order to determine the expected costs of maintenance, renewal theory can be applied when we can identify renewals that bring a component back into the as-good-as-new condition. This paper presents useful computational techniques to determine the probabilistic characteristics of a ren...