Forecasting counting and time statistics of compound Cox processes: a focus on intensity phase type process, deletions and simultaneous events

  • Paula R. BouzasEmail author
  • Nuria Ruiz-Fuentes
  • Carmen Montes-Gijón
  • Juan Eloy Ruiz-Castro
Regular Article


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.


Compound Cox process Estimation Principal components prediction Phase type process 

Mathematics Subject Classification

60G25 60G51 60G55 62M20 62M99 90C15 



This work was supported by Ministerio de Economía y Competitividad (project MTM2013-47929-P) and Consejería de Innovación de la Junta de Andalucía (Grants FQM-307 and FQM-246), all in Spain.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Paula R. Bouzas
    • 1
    Email author
  • Nuria Ruiz-Fuentes
    • 2
  • Carmen Montes-Gijón
    • 1
  • Juan Eloy Ruiz-Castro
    • 1
  1. 1.Department of Statistics and Operations ResearchUniversity of GranadaGranadaSpain
  2. 2.Department of Statistics and Operations ResearchUniversity of JaénJaénSpain

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