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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