Biomedical Engineering Reference
In-Depth Information
2 Model, with df g(;). Let g
1
and g
2
be the first and second marginal density
of g, respectively. Moreover, assume that
(S1) g
1
and g
2
are continuous, with g
1
(x) + g
2
(x) > 0 8 x 2 [0;M];
(S2) g(;) is continuous, with uniformly bounded partial derivatives, except
at a finite number of points, where left and right (partial) derivatives
exist;
(S3) PfY
2
Y
1
< g = 0 for some with 0 < 1=2M, so g does not have
mass close to the diagonal.
Then at each point t
o
2 (0;M)
n
1=3
f2a(t
o
)=f
o
(t
o
)g
1=3
f F(t
o
) F
o
(t
o
)g
D
! 2Z
;
where Z
is the last time where standard two-sided Brownian motion minus
the parabola y(t) = t
2
reaches its maximum, and
a(t
o
) =
g
1
(t
o
)
F
o
(t
o
)
g
2
(t
o
)
1 F
o
(t
o
)
;
+ k
1
(t
o
) + k
2
(t
o
) +
Z
M
Z
v
g(u;v)
F
o
(v) F
o
(u)
du:
Conjecture (G&W (1992, p. 108)): Suppose that F and F
U;V
have con-
tinuous derivatives, with their densities f(x
o
) > 0 and g(x
o
;x
o
) > 0. Then
(n ln n)
1=3
g(u;v)
F
o
(v) F
o
(u)
dv and k
2
(v) =
k
1
(u) =
u
0
F(x
0
)F(x
0
)
f
3
4
(f(x
0
))
2
=g(x
0
;x
0
)g
1=3
n
D
! 2Z
; where Z
is defined as in Theorem
3.
2.2.3
Various Models for MIC Data
1. The Mixed IC (MIC) Model I (Yu (1996) and (2000)). Assume
8
<
(U;V ]
if T 2 (U;V ];
(1) I =
where (U;V ) is an extended random
:
fTg otherwise ;
vector, that is, U may take value 1 (leading to left-censored
observations) and V may take value 1 (leading to right-censored
observations);
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