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

• Published in: Statistical papers (Berlin, Germany) Vol. 62; no. 1; pp. 235 - 265
• Main Authors: Bouzas, Paula R., Ruiz-Fuentes, Nuria, Montes-Gijón, Carmen, Ruiz-Castro, Juan Eloy
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-02-2021; Springer Nature B.V
• Subjects: Computer simulation; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Finance; Insurance; Management; Mathematical models; Mathematics and Statistics; Operations Research/Decision Theory; Prediction models; Probability Theory and Stochastic Processes; Regular Article; Statistics; Statistics for Business; Stochastic processes; 62M99; 60G55; Phase type process; Estimation; 60G51; Principal components prediction; 90C15; Compound Cox process; 60G25; 62M20
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-019-01092-0

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.