Biomedical Engineering Reference
In-Depth Information
^
n
and
The
maximum
likelihood
estimator
for and S
0
are
Q
j:t
j
t
e
exp(
^
n
j
)
that maximize
n
h
i
X
P
j:t
j
C
i
j
+Z
0
i
)
l(;) =
i
log
1 exp(e
i=1
9
=
X
j:t
j
C
i
(1
i
) exp(
j
+ Z
0
i
)
:
;
The
^
n
and
^
n
can be obtained by applying the Newton{Raphson algorithm.
To this end, we need the rst and second derivatives of l(;) which are not
shown here. Huang (1996) provided an alternative two-step algorithm. Both
approaches works well; however, when the number of distinct observation time
points or the dimension of is large, unstable estimation problems may arise
in addition to the intensive computation.
For continuous S
0
(t) and bounded Z
i
's, under some regularity conditions,
Huang (1996) shows that both
^
n
and S
n
(t) are consistent. However, the
convergence rate of the MLE is only n
1=3
, which is slower than the usual n
1=2
convergence rate. Specifically, under some regularity conditions, we have
d((
^
n
;
^
n
); (
0
;
0
)) = O
p
(n
1=3
);
where
0
and
0
are the true values of and , and d is the distance dened
on R
p
with
d((
1
;
1
); (
2
;
2
)) = j
1
2
j + jj
1
2
jj
2
:
Here, p is the dimension of ,
is the class of increasing functions that
bounded away from 0 and 1, and j
1
2
j is the l
2
norm, that is, the Euclidean
distance on R
p
and jj
1
2
jj is the L
2
norm dened by jjfjj
2
= (
R
f
2
dP)
1=2
with respect to the probability measure P.
The overall slow rate of convergence is dominated by
^
n
because the con-
vergence rate of
n
can still achieve
p
n. As shown in Huang (1996),
p
n(
^
n
0
) !
d
N(0;
1
);
Search WWH ::
Custom Search