Biomedical Engineering Reference
In-Depth Information
^
n
and
H
n
with
maximum likelihood estimators of and H
0
are dened as
H
n
(C
i
) = h
i
that maximize
X
f
i
log[1 exp((h
i
+ Z
0
i
))] (1
i
)(h
i
+ Z
0
i
)g
(; h) =
i=1
subject to h
1
h
2
h
n
, where h = (h
1
;:::;h
n
). We propose a two-
stage procedure to determine
^
n
and
H
n
. One can apply the following iterative
algorithm.
Step 1. Select initial estimates
(0
n
and H
(0
n
.
Step 2. At the l-th iteration, let =
(l1)
and apply the ICM algorithm
n
with H
(l1)
n
being the initial estimate to obtain H
(l
n
.
Step 3. Fix H = H
(l
n
and solve the score equation using an iterative algorithm
with
(l1)
n
being the initial estimate to obtain
(l
n
.
Step 4. Check if k
(l)
n
(l1
n
k
2
+ kH
(l
n
H
(l1
n
k
2
" for a given ". If
so, set
^
n
=
(l
n
and H
n
= H
(l
n
and stop. Otherwise, go back to Step 2.
Details of the ICM algorithm can be found in Groeneboom and Wellner
(1992) and Sun (2006). Under some regularity conditions, we proved that both
^
n
and H
n
are consistent. More specifically, we have
(
^
n
; H
n
); (
0
;H
0
)
= O
p
(n
1=3
) :
d
p
n(
^
n
0
) converges to a normal distribution with mean 0 and the variance-
covariance matrix achieves the information lower bound, which indicates that
^
n
is an ecient estimator.
4.5
Bivariate Current Status Data with Proportional
Odds Model
All the analyses described in the previous sections are for the cases in which
only one event is of interest. In this section we discuss regression analysis of
Search WWH ::
Custom Search