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A Review on Functional Data Analysis for Cox processes

217

intensity process characterizes the Cox process and its statistics. If the mean of

the Cox process is absolutely continuous, it is the integral of the intensity and

thus, is also a stochastic process. Indeed, the Cox process can be defined by

its mean process, as in Serfozo (1972), and a number of statistics can also be

characterized by the mean process. In brief, the intensity and the mean are both

stochastic processes which determine the Cox process. Therefore, by inferring

one of these processes, the target process and its statistics will be inferred as

well.

Functional data analysis permits the estimation of a stochastic process and

ensuing forecast by means of principal component prediction models under both

weak and usual conditions from observed sample paths of the process (Valder-

rama et al., 2000). Consequently, no structure or given distribution is assumed,

and thus, this technique is suitable for the development of an alternative infer-

ence method for the Cox process. As stated by Biau et al. (2013), despite the

large body of research on functional data analysis, relatively few attempts have

been made to connect it with stochastic processes. The present paper, then, will

review the application of functional data analysis to the inference of the intensity

and the mean processes without previous assumptions, employing only the raw

data observed from the Cox process. In addition, this review will provide the

estimation and prediction of a number of statistics of the Cox process and will

present a Cox process goodnes-of-fit test.

The following section contains the definitions of the Cox and compound Cox

process and the statistics that will be employed in the subsequent sections. The

inference for the mean process is presented in Section 3, where, additionally, the

application of this inference to the compound Cox process is derived. Similarly,

Section 4 presents the inference method for the intensity process and its conse-

quences. As a result of this method, a Cox process goodness-of-fit test can be

developed. The hypothesis test is described and is illustrated with an example.

Finally, the conclusions are drawn in the last section, particularly in relation to

the potential future uses of the Cox process.

2. Cox and compound Cox processes

This section presents the background necessary for the rest of the paper.

Further knowledge about other statistics or extensions of the Cox process (CP)

can be found in the references within the text.

2.1. Cox process

According to the classical approach, a CP

{

N

(

t

);

t

t

0

}

with intensity pro-

cess

{

λ

(

t, x

(

t

));

t

t

0

}

is defined as a conditional Poisson process with inten-

sity process

{

λ

(

t, x

(

t

));

t

t

0

}

given the information process

{

x

(

t

);

t

t

0

}

.

If

the mean, Λ(

t, x

(

t

)), of the CP is absolutely continuous, then Λ(

t, x

(

t

)) =