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Modeling Periodicity or Almost Periodicity in an Inhomogeneous Gamma Point Process


We propose a model for the analysis of an inhomogeneous Gamma processes (IGP) with a periodic or almost periodic baseline intensity function and shape parameter greater than zero. The IGP is a generalization of both the nonhomogeneous Poisson process and the renewal process with independent and identically distributed \Gamma(\kappa,\;\lambda) interarrival times, where \kappa>0 is the shape parameter and \lambda>0 is the scale parameter. The model deals with point events which are randomly spaced and show a pattern of periodicity or almost periodicity, such as the arrival times of customers to a store or the occurrence times of financial transactions. The concept of almost periodicity is described, and the purely periodic baseline intensity function is a particular case of the almost periodic baseline intensity function. We model the baseline intensity function as a sum of cosinusoidal functions plus a baseline constant. Given the number of periodic components K and shape parameter \kappa>0, we propose a mostly Bartlett periodogram based approach of constructing consistent frequency, amplitude, phase, and baseline constant estimators. Two methods of prediction of the next occurrence are also discussed. In practice, the values of K and \kappa are unknown, and we suggest the AIC and the BIC model selection criteria for determining K\in\{0\}\cup\mathbb{N} and \kappa\in\mathbb{N}. A simulation study is conducted to support theoretical results. A real data example is used to illustrate the theoretical results of our model. .

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