The intensity
λ
{\displaystyle \lambda }
of a counting process is a measure of the rate of change of its predictable part. If a stochastic process
{
N
(
t
)
,
t
≥
0
}
{\displaystyle \{N(t),t\geq 0\}}
is a counting process, then it is a submartingale , and in particular its Doob-Meyer decomposition is
N
(
t
)
=
M
(
t
)
+
Λ
(
t
)
{\displaystyle N(t)=M(t)+\Lambda (t)}
where
M
(
t
)
{\displaystyle M(t)}
is a martingale and
Λ
(
t
)
{\displaystyle \Lambda (t)}
is a predictable increasing process.
Λ
(
t
)
{\displaystyle \Lambda (t)}
is called the cumulative intensity of
N
(
t
)
{\displaystyle N(t)}
and it is related to
λ
{\displaystyle \lambda }
by
Λ
(
t
)
=
∫
0
t
λ
(
s
)
d
s
{\displaystyle \Lambda (t)=\int _{0}^{t}\lambda (s)ds}
.
Definition [ edit ]
Given probability space
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
and a counting process
{
N
(
t
)
,
t
≥
0
}
{\displaystyle \{N(t),t\geq 0\}}
which is adapted to the filtration
{
F
t
,
t
≥
0
}
{\displaystyle \{{\mathcal {F}}_{t},t\geq 0\}}
, the intensity of
N
{\displaystyle N}
is the process
{
λ
(
t
)
,
t
≥
0
}
{\displaystyle \{\lambda (t),t\geq 0\}}
defined by the following limit:
λ
(
t
)
=
lim
h
↓
0
1
h
E
[
N
(
t
+
h
)
−
N
(
t
)
|
F
t
]
{\displaystyle \lambda (t)=\lim _{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}_{t}]}
.
The right-continuity property of counting processes allows us to take this limit from the right.[1]
Estimation [ edit ]
In statistical learning , the variation between
λ
{\displaystyle \lambda }
and its estimator
λ
^
{\displaystyle {\hat {\lambda }}}
can be bounded with the use of oracle inequalities.
If a counting process
N
(
t
)
{\displaystyle N(t)}
is restricted to
t
∈
[
0
,
1
]
{\displaystyle t\in [0,1]}
and
n
{\displaystyle n}
i.i.d. copies are observed on that interval,
N
1
,
N
2
,
…
,
N
n
{\displaystyle N_{1},N_{2},\ldots ,N_{n}}
, then the least squares functional for the intensity is
R
n
(
λ
)
=
∫
0
1
λ
(
t
)
2
d
t
−
2
n
∑
i
=
1
n
∫
0
1
λ
(
t
)
d
N
i
(
t
)
{\displaystyle R_{n}(\lambda )=\int _{0}^{1}\lambda (t)^{2}dt-{\frac {2}{n}}\sum _{i=1}^{n}\int _{0}^{1}\lambda (t)dN_{i}(t)}
which involves an Ito integral . If the assumption is made that
λ
(
t
)
{\displaystyle \lambda (t)}
is piecewise constant on
[
0
,
1
]
{\displaystyle [0,1]}
, i.e. it depends on a vector of constants
β
=
(
β
1
,
β
2
,
…
,
β
m
)
∈
R
+
m
{\displaystyle \beta =(\beta _{1},\beta _{2},\ldots ,\beta _{m})\in \mathbb {R} _{+}^{m}}
and can be written
λ
β
=
∑
j
=
1
m
β
j
λ
j
,
m
,
λ
j
,
m
=
m
1
(
j
−
1
m
,
j
m
]
{\displaystyle \lambda _{\beta }=\sum _{j=1}^{m}\beta _{j}\lambda _{j,m},\;\;\;\;\;\;\lambda _{j,m}={\sqrt {m}}\mathbf {1} _{({\frac {j-1}{m}},{\frac {j}{m}}]}}
,
where the
λ
j
,
m
{\displaystyle \lambda _{j,m}}
have a factor of
m
{\displaystyle {\sqrt {m}}}
so that they are orthonormal under the standard
L
2
{\displaystyle L^{2}}
norm, then by choosing appropriate data-driven weights
w
^
j
{\displaystyle {\hat {w}}_{j}}
which depend on a parameter
x
>
0
{\displaystyle x>0}
and introducing the weighted norm
‖
β
‖
w
^
=
∑
j
=
2
m
w
^
j
|
β
j
−
β
j
−
1
|
{\displaystyle \|\beta \|_{\hat {w}}=\sum _{j=2}^{m}{\hat {w}}_{j}|\beta _{j}-\beta _{j-1}|}
,
the estimator for
β
{\displaystyle \beta }
can be given:
β
^
=
arg
min
β
∈
R
+
m
{
R
n
(
λ
β
)
+
‖
β
‖
w
^
}
{\displaystyle {\hat {\beta }}=\arg \min _{\beta \in \mathbb {R} _{+}^{m}}\left\{R_{n}(\lambda _{\beta })+\|\beta \|_{\hat {w}}\right\}}
.
Then, the estimator
λ
^
{\displaystyle {\hat {\lambda }}}
is just
λ
β
^
{\displaystyle \lambda _{\hat {\beta }}}
. With these preliminaries, an oracle inequality bounding the
L
2
{\displaystyle L^{2}}
norm
‖
λ
^
−
λ
‖
{\displaystyle \|{\hat {\lambda }}-\lambda \|}
is as follows: for appropriate choice of
w
^
j
(
x
)
{\displaystyle {\hat {w}}_{j}(x)}
,
‖
λ
^
−
λ
‖
2
≤
inf
β
∈
R
+
m
{
‖
λ
β
−
λ
‖
2
+
2
‖
β
‖
w
^
}
{\displaystyle \|{\hat {\lambda }}-\lambda \|^{2}\leq \inf _{\beta \in \mathbb {R} _{+}^{m}}\left\{\|\lambda _{\beta }-\lambda \|^{2}+2\|\beta \|_{\hat {w}}\right\}}
with probability greater than or equal to
1
−
12.85
e
−
x
{\displaystyle 1-12.85e^{-x}}
.[2]
References [ edit ]