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216

P. R. Bouzas, N. Ruiz-Fuentes

studying the occurrence of storms and the number of cells in an area. Erlang

(1909) also obtained the Poisson distribution for the number of phone calls arriv-

ing at a telephone office. Bateman (1910) examined the Rutherford and Geiger

problem of counting particles. Thereafter, increasingly complex approaches to

counting processes emerged in different fields including physics, ecology, astron-

omy and economics. Moyal (1962) provided the mathematical tools to address

the counting processes using finite-dimensional distributions and characteristic

functionals. At the same time, Cox (1955) discussed important issues about the

counting processes and several models were defined including the doubly stochas-

tic Poisson process. This process, which was a generalization of all the former

ones, is now known as Cox process. Since then, the literature about counting

process and the Cox process in particular has grown substantially.

The novelty of the Cox process was its doubly stochastic nature as the inten-

sity is a stochastic process itself. This fact confers the Cox process an important

level of flexibility which allows it to model a wide range of real phenomena. The

counting of the number of earthquakes is an example of this process as described

in Ogata (1998). Many other authors have studied the Cox process, includ-

ing Br`emaud (1981), Snyder and Miller (1991), Daley and Vere-Jones (2002),

Møller and Waagepetersen (2004), Wu et al. (2013) and Biau et al. (2013). Like

the Poisson process, the Cox process can be generalized in different ways. The

compound Cox process is a Cox process with a random variable associated to

each point called the mark of the point. Continuing with the example of the

number of earthquakes, the marks can indicate the intensities or severity of the

earthquakes. Further examples are provided by Economou (2003), Garc´ıa et al.

(2004), Barta et al. (2005) or Lin and Pavlova (2006). The Cox process can be

extended to the multidimensional case (multichannel or space-time ones must be

highlighted). Examples of the multidimensional Cox process are studied in Haas

and Shapiro (2002), Faenza et al. (2008), Renshaw et al. (2009), Myllymaki and

Pettinen (2010), Fern´andez-Alcal´a et al. (2012), etc.

A review of the literature on Cox processes and their generalizations re-

veals different inference techniques. These techniques, however, usually assume

a given structure of the process, such as a fixed stochastic structure of the inten-

sity process, which is not completely which may not be completely convenient;

moreover, the purpose of those techniques is normally one of estimation rather

than forecasting. On the other hand, the use of Cox processes to model and

forecast the data of a real phenomena is still sparse. As discussed by Snyder

and Miller (1991), a possible reason for this scarcity lies in the fact that the

statistics of the Cox process can be easy to define using the conditioning method

but difficult, if not impossible, to tackle in particular examples.

The Cox process can be defined from several different perspectives including

martingale theory, survival theory and event history theory. This paper will

use the classical approach given by Snyder and Miller (1991). In all cases, the