Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions a
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Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events Paula R. Bouzas1 · Nuria Ruiz-Fuentes2 · Carmen Montes-Gijón1 · Juan Eloy Ruiz-Castro1 Received: 26 February 2018 / Revised: 24 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Compound Cox processes (CCP) are flexible marked point processes due to the stochastic nature of their intensity. This paper states closed-form expressions of their counting and time statistics in terms of the intensity and of the mean processes. They are forecast by means of principal components prediction models applied to the mean process in order to reach attainable results. A proposition proves that only weak restrictions are needed to estimate the probability of a new occurrence. Additionally, the phase type process is introduced, which important feature is that its marginal distributions are phase type with random parameters. Since any non-negative variable can be approximated by a phase-type distribution, the new stochastic process is proposed to model the intensity process of any point process. The CCP with this type of intensity provides an especially general model. Several simulations and the corresponding study of the estimation errors illustrate the results and their accuracy. Finally, an application to real data is performed; extreme temperatures in the South of Spain are modeled by a CPP and forecast. Keywords Compound Cox process · Estimation · Principal components prediction · Phase type process Mathematics Subject Classification 60G25 · 60G51 · 60G55 · 62M20 · 62M99 · 90C15
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Paula R. Bouzas [email protected]
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Department of Statistics and Operations Research, University of Granada, Granada, Spain
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Department of Statistics and Operations Research, University of Jaén, Jaén, Spain
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1 Introduction The compound Cox process (CCP) is the natural extension of the compound Poisson process and the Cox process (CP) since it is a Poisson process with stochastic intensity and also a marked point process. More precisely, a CCP is a marked point process in which the point process is a CP and the marks of the events are independent and identically distributed as well as independent of the CP Snyder and Miller (1991). The CCP has been used in several fields including geology, as in the classical application to earthquakes by Ogata (1998) or Gospodinov and Rotondi (2001); demography (Economou 2003); risk theory (Lin and Pavlova 2006); econometrics (Chertok et al. 2016); astrophysics (Si 2001), etc. However, although this is quite a general and flexible model, it is rare to find systematic studies of this type of process. On the contrary, studies on CP or CCP often use a limited variety of examples where the intensity or the mean process has an assumed stochastic structure that can fit the phenomena under consideration (Bouzas et al. 2002; Genaro and Simonis 2015). The main reason may lay in the difficulty of estimating the parameters of
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