A Review on Functional Data Analysis for Cox processes

219

It can be very important to study the occurrences of a CCP whose marks are

in a given subset of the mark space. This is the counting process,

{

N

(

t, B

);

t

≥

t

0

}

, of the points from the former CCP whose marks are in

B

⊆ U

. The repre-

sentation theorems presented in Bouzas et al. (2007) prove that

N

(

t, B

) is also

a CP for any

B

with the following intensity process given a non-denumerable

mark space

λ

(

t, x

(

t

))

$

B

P

u

(

dU

)

,

and the following mean process

Λ(

t, x

(

t

))

$

B

P

u

(

dU

)

.

The analogous result for a denumerable space is given in the reference made

above. The main conclusion of this result is that if the study interest lies not

on all the points but on those with specific marks (marks in

B

), it is possible to

take the process

N

(

t, B

) and treat it like the CP that it is. However, in order

to study this process, it is not necessary to extract the points from the original

CCP but to study the intensity or the mean regardless of the marks and then

multiplying by the constant in terms of the chosen subset,

!

B

P

u

(

dU

).

Studying the process

N

(

t, B

) is not restrictive as

B

could be chosen as

U

if applicable. Thus, the next statistics under study are the corresponding to

N

(

t, B

) in the case of a non-denumerable mark space.

Using the representation theorems, the expressions of the counting statistics

are presented below.

•

Probability mass function of

N

(

t, B

)

P

[

N

(

t, B

) =

n

] =

E

x

*

1

n

!

#

Λ(

t, x

(

t

))

$

B

P

u

(

dU

)

%

n

exp

)

−

Λ(

t, x

(

t

))

$

B

P

u

(

dU

)

.-

=

1

n

!

G

Λ(

t,x

(

t

))

!

B

P

u

(

dU

)

(

−

1)

•

Mean of

N

(

t, B

)

E

[

N

(

t, B

)] =

E

[Λ(

t, x

(

t

))]

$

B

P

u

(

dU

)

•

Mode of

N

(

t, B

)

The most probable number of occurrences in

B

until a certain time point