The Exponential Power-Law Process for the Nonhomogeneous Poisson Process and Bathtub Data

Recently, Bar-Lev, Bshouty and Van der Duyn Schouten [Math. Methods Stat. 25 (2016) 79-980] developed a systematic method, called operator-based intensity function, for constructing huge classes of nonmonotonic intensity functions (convex or concave) for the nonhomogeneous Poisson process, all of which are suitable for modeling bathtub data. Each class is parametrized by several parameters (as scale and shape parameters) in addition to the operator index parameter n . For the sake of demonstration only, we focus in this paper on a special subclass called the exponential power law process (EXPLP(n)) whose base function is the intensity function of the power-law process. We describe various properties of such a subclass and use one of its special case, namely EXPLP(1) intensity function, to analyze failure data which lack monotonicity. Maximum likelihood estimation of the parameters involved and relevant functions thereof is discussed with respect various aspects as existence, uniqueness, asymptotic behavior and statistical inference facets. Using two real datasets from the literature we provide evidence that the EXPLP(1) intensity function is well suited to analyze data which exhibit a bathtub behavior.