Biomedical Engineering Reference
In-Depth Information
The proposed estimator
^
is the solution to the score equation
@
^
L()=@ = 0. Estimates can be obtained by using the Newton-Raphson
iterative procedure. The following theorem is due to Weaver and Zhou
14
.
Theorem 2: Under some regularity conditions the proposed estimator
^
is asymptotic normally distributed. That is
p
N(
^
)!
D
N(0; )
as N!1;
where
X
K
k
= I
1
() +
k
V
+
k
0
V
I
1
()
k
()I
1
()
k=1
where
K
X
@
2
log f
(YjX)
@@
0
@
2
log f
(YjX)
@@
0
I() =
0
V
E
k
V
E
k
k=1
K
X
@
2
log f
(Y )
@@
0
[
k
(1
0
V
)
k
V
]E
k
k=1
(
)
X
K
k
() = Var
XjY2C
k
[
k
1
(1
0
V
)
k
1
V
]E
XjY2C
k
[M
X
(Y ; )]
k
1
=1
and
M
X
(Y ; ) =
@f
(YjX)=@
@f
(Y )=@
( f
(Y ))
2
f
(YjX):
f
(Y )
A consistent estimator for can be constructed using sample quantities.
Specically, dene
K
X
(N
k
=N)
2
(n
k
+ n
0;k
)=N
^
= I
1
(
^
) +
I
1
(
^
)
^
k
(
^
) I
1
(
^
);
k=1
where
(
)
@
2
^
L()
@@
0
;
X
K
n
V ;k
N
I() =
1
N
M
X
i
;l
()
^
k
() =
Var
(X
i
:i2V
k
)
l=1
with
8
<
9
=
:
@f
(Y
j
jX
i
)
f(Y
j
jX
i
)
@f
(Y
j
)
X
M
X
i
;l
() =
@
f
(Y
j
)
@
=n
V ;l
:
h
f
(Y
j
)
i
2
;
j2V
l
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