Biomedical Engineering Reference
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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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