We introduce two classes of point processes: a fractional non-homogeneous Poisson process order k and Pólya-Aeppli k. characterize these processes by deriving their non-local governing equations. further study the covariance structure investigate long-range dependence property.

We present supporting material in this online supplement to the main paper. In §2 we define the CU and Lewis KS tests of an NHPP that we consider in the main paper. In §3, we illustrate how departures from the Poisson property can have a strong impact upon performance via simulation experiments. In §?? we give the results from addtional experiments on rounding and un-rounding. In §5 we provide ...

The repairable system suffers various type of failure and each failure type has different repair cost. Assume the failure process of the system as Non-homogenous Poisson Process (NHPP). The system is replaced after it experienced a predetermined number of minimal repairs. Considering finite time horizon, the paper proposes a replacement model for the system. It firstly proves that the failure p...

We study Manin triples for a reductive Lie algebra, g. First, we generalize results of E. Karolinsky, on the classification of Lagrangian subalgebras (cf. KAROLINSKY E., A Classification of Poisson homogeneous spaces of a compact Poisson Lie group, Dokl. Ak. Nauk, 359 (1998), 13-15). Then we show that, if g is non commutative, one can attach , to each Manin triple in g , another one for a stric...

A system subject to a point process of shocks is considered. Shocks occur in accordance with a non-homogeneous Poisson process. Different criterions of system failures are discussed in a homogeneous case. Two natural settings are analyzed. Heterogeneity is modeled by an unobserved univariate random variable (frailty). It is shown that reliability (safety) analysis for a heterogeneous case can d...

Consider the Boolean model in IR, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see e.g. Section 9.5.3 in Cressie (1993)) that Laslett’s rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity λ. In the present paper, we give a simple proof o...

The paper concentrates on modeling the data that can be described by a homogeneous or non-homogeneous Poisson process. The goal is to decide whether the intensity of the process is constant or not. In technical practice, e.g., it means to decide whether the reliability of the system remains the same or if it is improving or deteriorating. We assume two situations. First, when only the counts of...

Let N , N1 and N2 be point processes such that N1 is obtained from N by homogeneous independent thinning and N2 = N −N1. We give a new elementary proof that N1 and N2 are independent if and only if N is a Poisson point process. We also present an application of this result to test if a homogeneous point process is a Poisson point process.

Nonhomogeneous Poisson process (NHPP) also known as Weibull process with power law, has been widely used in modeling hardware reliability growth and detecting software failures. Although statistical inferences on theWeibull process have been studied extensively by various authors, relevant discussions on predictive analysis are scattered in the literature. It is well known that the predictive a...