Biomedical Engineering Reference
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paper. It shows that under mild conditions, namely that F0i(∞))
0i
(t
0
) 2 (0;F
0i
(1))
for i = 1; 2; and that the F
0i
's and G are continuously dierentiable in a
neighborhood of t
0
with the derivatives at t
0
, ff
0i
(t
0
)g
i=1
and g(t
0
), being
positive, for any 2 (0; 1), there is a constant r > 0 such that
j F
+
(t) F
+
(t)j
v
n
(tt
0
)
sup
t2[t
0
r;t
0
+r]
= O
p
(1) ;
where v
n
(s) = n
1=3
1(jsj n
1=3
) + n
(1)=3
jtj
1(jsj > n
1=3
). Thus, the
local rate of F
+
, the MLE of the distribution function, is the same as in
the current status model (as the form of v
n
(s) for jsj n
1=3
shows), but
outside of the local n
1=3
neighborhood of t
0
, the normalization changes (as
the altered form of v
n
shows). This result leads to some crucial bounds that
are used in the proof of Theorem 4.17 of their paper, where it is shown that
given ;M
1
> 0, one can nd M;n
1
> 0 such that for each i,
!
n
1=3
j F
i
(s + n
1=3
h) F
0i
(s)j > M
P
sup
h2[M
1
;M
1
]
< ;
for all n > n
1
and s varying in a small neighborhood of t
0
. Groeneboom
et al. (2008b) make further inroads into the asymptotics; they determine the
pointwise limit distributions of the MLEs of the F
0i
's in terms of completely
new distributions, the characterizations of which, again, require much di-
cult work. Let W
1
and W
2
denote a couple of correlated Brownian motions
originating from 0 with mean 0 and covariances
E(W
j
(t) W
k
(s)) = (jsj^jtj) 1fst > 0g
jk
; s;t 2R; 1 j;k 2 ;
with
jk
= g(t
0
)
1
[1fj = kgF
0k
(t
0
) F
0k
(t
0
) F
0j
(t
0
)]. Note the multi-
nomial covariance structure of ; this is not surprising in the light of
the observation that the conditional distribution of given U = t
0
is
Multinomial(1;F
01
(t
0
);F
02
(t
0
);S(t
0
)). Consider the drifted Brownian mo-
tions (V
1
;V
2
) given by V
i
(t) = W
i
(t) + (f
0i
(t
0
)=2) t
2
for i = 1; 2. The limit
distribution of the MLEs can be described in terms of certain complex func-
tionals of the V
i
's that have self-induced characterizations. Before describing
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