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Bolet´ın de Estad´ıstica e Investigaci´on Operativa

Vol. 31, No. 3, Noviembre 2015, pp. 215-230

Estad´ıstica

A Review on Functional Data Analysis for Cox processes

Paula R. Bouzas

Departamento de Estad´ıstica e I.O.

Universidad de Granada

!

paula@ugr.es

Nuria Ruiz-Fuentes

Departamento de Estad´ıstica e I.O.

Universidad de Ja´en

!

nfuentes@ujaen.es

Abstract

Counting processes have evolved from the simple Poisson process into

complex models such as the Cox process. The key feature of the latter

is the doubly stochastic nature due to its stochastic intensity. The litera-

ture provides different inference techniques for the Cox process. However,

these techniques usually assume a given structure of the process and their

purpose is normally one of estimation rather than forecasting. This pa-

per reviews an alternative. Functional data analysis can be applied to the

intensity or the mean as they are processes themselves. Therefore, an in-

ference technique is derived for the Cox process from raw observed data

without assuming stochastic structure.

Keywords:

Cox process, functional data analysis, stochastic intensity,

inference

AMS Subject classifications:

60G25, 60G51, 60G55, 62M20, 62M99,

90C15

1. Introduction

A point process is a phenomena such that an event occurs repeatedly and

randomly in time or space. The counting of these events or points generates

the counting process. The first approach to the counting of events was made

by Sim`eon Denis Poisson (1781 - 1840), who published the finding of the Pois-

son distribution as the limit of a Binomial in Poisson (1837). Early discussions

about counting problems were made by Seidel (1876) and Abb`e (1879), while

c

2015 SEIO