Poissonova

Poissonova is a term used in probability theory to denote a generalized counting process that extends the classical Poisson process by allowing the arrival rate to be random and time-dependent. In many expositions, Poissonova is described as a pedagogical name for a Cox process, or doubly stochastic Poisson process.

Definition: Let Λ(t) be a nonnegative stochastic process adapted to a filtration. Given Λ, the counting process N(t) is Poisson with mean ∫0^t Λ(s) ds. This makes N(t) a Poisson process with stochastic intensity.

Properties: Mean of N(t) equals E[∫0^t Λ(s) ds], and Var(N(t)) equals E[∫0^t Λ(s) ds] plus Var(∫0^t Λ(s) ds). Hence displaying overdispersion relative to a fixed-rate Poisson process. If Λ is constant, Poissonova reduces to ordinary Poisson; if Λ has gamma distribution, the marginal distribution of N(t) is negative binomial.

Applications and methods: Poissonova models are used to describe irregular event arrivals with varying rate, such as in epidemiology, seismology, reliability, finance, and telecommunications. Inference can be done via Bayesian methods, EM algorithms, or simulation-based approaches; thinning and data augmentation are common computational techniques.

Etymology and status: The name Poissonova is not standardized in formal literature and often appears in textbooks or pedagogical materials as a friendly label for the class of doubly stochastic Poisson processes. More common terminology uses Cox processes or Poisson processes with stochastic intensity.