Biomedical Engineering Reference
In-Depth Information
where
0
is a d-dimensional regression parameter and
0
is the unspecified
baseline cumulative hazard function.
Denote the two censoring variables by U and V , where P(U V ) = 1. Let
G be the joint distribution function of (U;V ). Let
1
= 1
[TU]
;
2
= 1
[U<TV ]
and
3
= 1
1
2
. Assume that conditionally on Z, T is independent of
(U;V ), and that the joint distribution of (U;V ) and Z does not depend on
the parameters of interest. Then the density function of X (
1
;
2
;U;V;Z)
with respect to the product of the counting measure on f0; 1gf0; 1g and the
probability measure induced by the distribution of (Z;U;V ) is
(1 exp((u)e
0
z
))
1
p(x; ; )
=
(exp((u)e
0
z
) exp((v)e
0
z
))
2
(exp((v)e
0
z
)
3
:
Let = log . We reparameterize this log-likelihood in terms of (;). The
resulting log-likelihood for a sample of size 1 is, up to an additive term not
dependent on (;),
`(x; ;) = log p(x; ;):
Let
X
= (X
1
;X
2
; ;X
n
) with X
i
= (
1i
;
2i
;U
i
;V
i
;Z
i
), for 1 i n being
a random sample with the same distribution as X = (
1
;
2
;U;V;Z). The
log-likelihood for this random sample is
X
`
n
(;) =
`(X
i
; ;):
i=1
Because is a nondecreasing function, it is desirable to restrict its estimator
to be also nondecreasing. Therefore, we seek an estimate of in the space
M
n
M
n
(D
n
;K
n
;m) dened below.
Let b
n
= (b
1
;:::;b
q
n
) be the basis of B-splines defined earlier. The mono-
tone polynomial spline space is defined to be
M
n
(D
n
;K
n
;m) =
n
o
q
X
(9.14)
n
:
n
(t) =
j
b
j
(t) 2S
n
(D
n
;K
n
;m); 2 B
n
;t 2 [a;b]
:
j=1
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