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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