Operator-based intensity functions for the nonhomogeneous Poisson process

Abstract

A nonhomogeneous Poisson process (NHPP) plays an important role in a variety of applications as reliability of repairable systems, software reliability and actuarial studies. An NHPP is characterized by its intensity function m(t), which provides information on the time-dependent nature of the reliability of the system. Various intensity functions, which describe different behavior (from reliability decay to reliability growth along with monotonicity, convexity or concavity), have been suggested for NHPP’s for modeling repairable systems. Perhaps one of the most frequently utilized NHPP is the power lawprocess (PLP) in which m(t) is a power function of t. Inthis studywe present a general method for constructing new intensity functions for NHPP’s yielding new classes of NHPP’s. This method utilizes certain operators Ln, n ∈ N0, acting on some suitable functions L0 = f (termed base functions). We call these classesOBIF’s (operator-based intensity functions). These classes are represented in terms of three parameters of which one is an indexing parameter n ∈ N0 and two others are scale and shape parameters. The fact that n ∈ N0 is also a parameter provides a flexibility in the choice of the appropriate statistical model for NHPP’s data. In particular, we consider the exponential operator acting on the PLP intensity function f and realize that Ln’s, n ≥ 2, inherit properties similar to those of L1 (convexity and concavity) and thus are suitable for modelling bathtub data. We also consider a more comprehensive treatment of OBIF classes where both, the operator and base functions, are general. All of the introduced operators are demonstrated with illustrative examples.