Biomedical Engineering Reference
In-Depth Information
6. The Mixed Case IC Model (Schick and Yu, 2000). The model is a mixture
of various Case k models. In particular, assume
8
<
(1;Y
K;1
]
if if Y
K;1
(1) I =
(Y
K;j1
;Y
K;j
]
if Y
K;j1
< T Y
K;j
;j 2f2;::;Kg
:
(Y
K;K
;1) if T > Y
K;K
;
(2) conditional on K = k, T ? (Y
k;1
;:::;Y
k;k
);
(3) F
K;Y
is noninformative, where Y = fY
kj
: j 2f1;:::;kg; k 1g.
8
<
f
Y
K;1
(k;u)
if I = (1;u]
P
j=1
f
Y
K;j1
;Y
K;j
(u;v)
Note: G(I) =
In the mixed
if I = (u;v]
:
f
Y
K;K
(v)
if I = (v;1):
case model, even though it is assumed the conditional independence,
it is not assumed that T ? C, where C = (K; Y). The asymptotic
properties of the NPMLE
F of F under various IC models have been
established.
Theorem 1 (Schick and Yu, 2000) Assume that (L;R) satises the Mixed
Case Model with E(K) < 1. Then
R
j F Fjd ! 0 almost surely, where
= F
R
+ F
L
.
Theorem 2 (Yu et al. (1998b),Yu et al. (1998a)) Under the Case 1 or
Case 2 IC Model, if there are only k + 1 innermost intervals with right
endpoints bi
i
for each sample size, F is strictly increasing on bi
i
's, then
0
1
F(b
1
) F(b
1
)
F(b
k
) F(b
k
))
@
A
p
n
D
! N(0; ) as n ! 1, where the k k matrix
can be estimated by the inverse of the empirical Fisher information matrix.
Theorem 3 (Groeneboom, 1996). Let F be continuous with a bounded deriva-
tive f on [0;M], satisfying f(x) c > 0; x 2 (0;M); for some constant
c > 0. Let (Y
1
;Y
2
) be the two continuous random inspection times in the Case
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