Biomedical Engineering Reference
In-Depth Information
xed , one has
g)1
3
n
k
3
Y
Y
Y
n
k
k
L
2
(;fp
ki
p
ki
:
i=1
k=0
k=1
fp
ki
g;k = 0;:::;K;i = 1;:::;n
k
can be searched by maximizingL
2
under
the following constraints:
"
#
X
K
X
n
k
X
K
X
n
k
p
ki
= 1;
p
ki
fPr(Y2C
s
jX
ki
)
s
g= 0;s = 1; 3
i=1
i=1
k=0
k=0
where p
ki
0. These constraints reect the properties of G
X
(x) as a dis-
crete distribution with support points at the observed X values. Using the
Lagrange multiplier argument to derive the maximum overfp
ki
g. Speci-
cally, write
!
1
K
X
X
n
k
H= logL
2
(;fp
ki
g) +
p
ki
k=0
i=1
"
#
X
K
X
n
k
n
1
p
ki
fPr(Y2C
1
jX
ki
)
1
g
k=0
i=1
"
#
X
K
X
n
k
n
3
p
ki
fPr(Y2C
3
jX
ki
)
3
g
i=1
k=0
where and (
1
;
3
)
0
s are Lagrange multipliers. Take derivatives with
respect to p
ki
, and setting
X
K
X
n
k
@H=@p
ki
= 0
and
p
ki
@H=@p
ki
= 0;
k=0
i=1
one has
1
n
1
= n; p
ki
=
1 +
1
fPr(Y2C
1
jX
ki
)
1
g+
3
fPr(Y2C
3
jX
ki
)
3
g
;
with restriction
K
X
X
n
k
1
n
Pr(Y2C
j
jX
ki
)
j
s=1
s
fPr(Y2C
s
jX
ki
)
s
g
= 0; for j = 1;:::;K1:
where n = n
0
+ n
1
+ n
3
. Let
1
=
1
P
K1
1 +
k=0
i=1
k
3
=
3
, k
i
= n
i
=n,
i = 0; 1; 3, = (
1
;
3
;
1
;
3
),
0
= (
0
;
0
). The log transformation of the
resulting prole likelihood function has the form
k
1
=
1
,
3
=
3
l() = logL
1
() + logL
2
(;) = l
1
() + l
2
(;);
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