Biomedical Engineering Reference
In-Depth Information
up to an additive term not involving F, is readily seen to be
X
L
n
(F)
=
[
i
log F(U
i
) + (1
i
) log(1 F(U
i
))]
i=1
X
=
[
(i)
log F(U
(i)
) + (1
(i)
) log(1 F(U
(i)
))] ;
i=1
where U
(i)
is the i-th smallest of the U(i)
j
's and
(i)
its corresponding indicator.
The LRS is then given by
LRS(
0
) = 2 [L
n
( F
n
) L
n
( F
n
)] ;
where F
n
is the NPMLE and F
n
the constrained MLE under the null hypothe-
sis F(t
0
) =
0
. Let F
n
(U
(i)
) = v
i
and F
n
(U
(i)
) = v
i
. It can then be shown, via
the Fenchel conditions that characterize the optimization problems involved
in finding the MLEs, that
X
fv
i
g
i=1
= arg
[
(i)
s
i
]
2
;
min
s
1
s
2
:::s
n
(3.3)
i=1
and that
X
fv
i
g
i=1
= arg
[
(i)
s
i
]
2
;
min
s
1
s
2
:::s
m
0
s
m+1
:::s
n
(3.4)
i=1
where U
(m)
t
0
U
(m+1)
. Thus, F
n
and F
n
are also solutions to least
squares problems. They are also extremely easy to compute using the PAV al-
gorithm, having nice geometrical characterizations as slopes of greatest convex
minorants. To describe these characterizations, we introduce some notation.
First, for a function g from an interval I toR, the greatest convex minorant
or GCM of g will denote the supremum of all convex functions that lie be-
low g. Note that the GCM is itself convex. Next, consider a set of points in
R
2
; f(x
0
;y
0
); (x
1
;y
1
);:::; (x
k
;y
k
)g, where x
0
= y
0
= 0 and x
0
< x
1
< ::: < x
k
.
Let P(x) be the left continuous function such that P(x
i
) = y
i
and P(x)
are constant on (x
i1
;x
i
). We denote the vector of slopes (left-derivatives) of
the GCM of P(x), at the points (x
1
;x
2
;:::;x
k
), by slogcmf(x
i
;y
i
)g
i=0
. The
GCM of P(x) is, of course, also the GCM of the function that one obtains
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