Poissonhypyt

Poissonhypyt is a theoretical construct in probability and statistics that describes a class of point processes extending the classical Poisson process by allowing the event rate to vary randomly over time. The name is used in a minority of literature and is sometimes treated as a synonym or alias for doubly stochastic Poisson processes, also known as Cox processes.

In the Poissonhypyt framework, the counting process N(t) is conditionally Poisson given a stochastic intensity Λ(t). Conditional on Λ(t), N(t) has increments that follow a Poisson distribution with mean ∫0^t Λ(s) ds. The randomness of Λ(t) induces overdispersion and temporal clustering in N(t). Λ(t) may be modeled as a positive diffusion or a Markov-modulated process, such as a Gamma process, a lognormal diffusion, or a discrete-time Markov chain (a Markov-modulated Poisson process).

Metrics and properties: The unconditional distribution of counts is typically not Poisson; the process exhibits overdispersion (variance exceeding the mean) and autocorrelation in counts. It is related to, but distinct from, Hawkes processes, which introduce self-excitation through feedback from past events rather than a latent intensity.

Estimation and inference: Common approaches include likelihood-based methods with data augmentation, Bayesian inference via MCMC or particle filtering, and moment-based estimators. Computational complexity can be high and identifiability may depend on the chosen form of Λ(t).

Applications: The Poissonhypyt model is used to describe bursty event data in telecommunications, finance (trade counts), seismology, epidemiology (case counts), and reliability engineering. It provides a flexible alternative to homogeneous Poisson and standard Hawkes models when overdispersion is present.

Status and terminology: Poissonhypyt is not universally standardized and is sometimes superseded by the broader term Cox process or doubly stochastic Poisson process. When encountered, it typically signals a modeling choice that treats the intensity as a random process rather than a fixed parameter.