Biomedical Engineering Reference
In-Depth Information
= 0,
Y
L(0;S
0
) =
fS
0
(`
i
) S
0
(r
i
)g:
(13.2)
i=1
Let 0 = t
0
< t
1
< < t
m
< t
m+1
1 denote the ordered ob-
served ends of the intervals (i.e., the ordered unique values in the set
f0;L
1
;R
1
;L
2
;R
2
;:::;L
n
;R
n
;1g). There is not a unique solution to S
0
that
maximizes the likelihood given in Equation ( 13.2); the solution to the likeli-
hood is only unique at t
j
, j = 0; 1;:::;m+ 1 (see Gentleman and Geyer, 1994;
Gentleman and Vandal, 2002). We call any survival function that maximizes
Equation ( 13.2) the nonparametric maximum likelihood estimate (NPMLE),
despite the fact that the NPMLE really represents a class of survival functions,
in which each member of the class has the same values at the observed ends
of the intervals (i.e., at the t
j
values). Let the NPMLE be denoted S
0
. The
NPMLE can be estimated by the E-M algorithm (Dempster et al., 1977), also
known in this context as the self-consistent algorithm (Turnbull, 1976; Gentle-
man and Geyer, 1994), but there have been many other algorithms that have
increased computational speed (see, for example, Wellner and Zhan, 1997;
Gentleman and Vandal, 2001; Groeneboom et al., 2008).
Then the ecient score statistic can be written as
@ logfL(;S
0
)g
@
X
U =
=
z
i
c
i
:
(13.3)
=0;S
0
= S
0
i=1
We can write the scores ci
i
as
S
0
(`
i
) S
0
(r
i
)
S
0
(`
i
) S
0
(r
i
)
c
i
= c(y
i
; S
0
) =
;
(13.4)
where
@ logfS(t; = z
0
i
;S
0
)g
@
S
0
(t) =
:
=0;S
0
= S
0
We write ci
i
= c(y
i
; S
0
) to emphasize that it is a function of yi
i
and S
0
, and
it acts like a ranking function.
When F is the extreme value distribution, the score test in this case is the
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