Biomedical Engineering Reference
In-Depth Information
This proposition was proved in Huang and Wellner (1997) and Huang and
Rossini (1997). It is the basis for the asymptotic normality and eciency of
b
n
(Theorem 9.2) and is needed in the proof of the consistency of the proposed
variance estimator.
9.4.2
Poisson Proportional Mean Model for Panel Count
Data
Let fN(t) : t 0g be a univariate counting process. K is the total num-
ber of observations on the counting process and
T
= (T
K;1
; ;T
K;K
) is a
sequence of random observation times with 0 < T
K;1
< < T
K;K
. The
counting process is only observed at those times with the cumulative events
denoted byN= fN(T
K;1
); ;N(T
K;K
)g. This type of data is referred to as
panel count data by Sun and Kalbfleisch (1995). Panel Count Data can be
regarded as a generalization of mixed-case interval-censored data to the sce-
nario that the underlying counting process is allowed to have multiple jumps.
Here we assume that (K;
T
) is conditionally independent ofNgiven a vec-
tor of covariates Z, and we denote the observed data consisting of indepen-
dent and identically distributed X
1
; ;X
n
, where X
i
= (K
i
;T
(i)
;N
(i)
;Z
i
)
with T
(i)
= (T
(i)
K
i
;1
; ;T
(i)
(i)
= (N
(i)
(T
(i)
K
i
;1
); ;N
(i)
(T
(i)
K
i
;K
i
) andN
K
i
;K
i
)), for
i = 1; ;n.
Panel count data arise in clinical trials, social demographic, and indus-
trial reliability studies. Sun and Wei (2000) and Zhang (2002) considered the
proportional mean model
(tjz) =
0
(t) exp(
0
0
z)
(9.16)
for
analyzing
panel
count
data
semiparametrically,
where
(tjz)
=
E(N(t)jZ = z) is the expected cumulative number of events observed at time
t, conditionally on Z with the true baseline mean function given by
0
(t).
Wellner and Zhang (2007) proposed a nonhomogeneous Poisson process with
the conditional mean function given by Equation (9.16) to study the MLE of
Search WWH ::
Custom Search