Biomedical Engineering Reference
In-Depth Information
0
= (
0
;
0
(t)). The log-likelihood for (; (t)) under the Poisson proportional
mean model is given by
h
N
i
;
X
K
X
(i)
K
i
;j
log
K
i
;j
+ N
(i)
K
i
;j
0
Z
i
exp(
0
Z
i
)
K
i
;j
`
n
(; ) =
i=1
j=1
where
(i)
K
i
;j
=N
(i)
(T
(i)
K
i
;j
) N
(i)
(T
(i)
N
K
i
;j1
)
and
K
i
;j
= (T
(i)
K
i
;j
) (T
(i)
K
i
;j1
);
for j = 1; 2; ;K.
To study the asymptotic properties of the MLE, Wellner and Zhang (2007)
defined the following L
2
-norm,
1=2
Z
j
1
2
j
2
+
j
1
(t)
2
(t)j
2
d
1
(t)
d(
1
;
2
) =
;
with
Z
X
k
X
1
(t) =
P(K = kjZ = z)
P(T
K;j
tjK = k;Z = z)dF(z):
R
d
k=1
j=1
They showed that the semiparametric MLE, b
n
= (
^
n
;
^
n
) converges to the
true parameters
0
= (
0
;
0
) (under some mild regularity conditions) in a
rate lower than n
1=2
, that is, d(b
n
;
0
) = O
p
(n
1=3
); however, the MLE of
0
is still semiparametrically ecient, that is,
p
n(
^
n
0
) !
d
N
0;I
1
(
0
)
with the Fisher information matrix given by
8
<
#
2
9
"
=
Z
E(Ze
0
0
Z
jK;T
K;j1
;T
K;j
)
E(e
0
0
Z
jK;T
K;j1
;T
K;j
)
0
(T
K;j
)e
0
0
Z
I(
0
) = E
:
:
;
The computation of the semiparametric MLE is very time consuming as
it requires the joint estimation of
0
and the infinite-dimensional parameter
0
. Although the Fisher information has a nice explicit form, there is no easy
method available to calculate the observed information.
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