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α σ ( j ) is the j th largest of m
where J , n
κ α σ ( j ) α σ ( j )
ω i J , n
κ α i α i ( α i ) (
=
,
,...,
)
i
1
2
m
.
(8) GIFPHAP w 1 2 ,...,α m )
w 1 P n
α σ ( 1 ) ρ
w 2 P n
α σ ( 2 ) ρ
=
κ α σ ( 1 ) α σ ( 1 )
κ α σ ( 2 ) α σ ( 2 )
w m P n
α σ ( m ) ρ
1
ρ
⊕···⊕
κ α σ ( m ) α σ ( m )
α σ ( j ) is the j th largest of
m , P n
κ α σ ( j ) α σ ( j )
κ α σ ( j ) + λ α σ ( j )
=
,
,...,
where
1, j
1
2
ω i P n
m
κ α i α i i ) (
i
=
1
,
2
,...,
m
)
.
(9) GIFPHAQ w 1 2 ,...,α m )
w 1 Q n
α σ ( 1 ) ρ
w 2 Q n
α σ ( 2 ) ρ
=
κ α σ ( 1 ) α σ ( 1 )
κ α σ ( 2 ) α σ ( 2 )
w m Q n
α σ ( m ) ρ
1
ρ
⊕···⊕
κ α σ ( m ) α σ ( m )
α σ ( j ) is the j th largest
m , Q n
κ α σ ( j ) α σ ( j )
where
κ α σ ( j ) + λ α σ ( j )
1, j
=
1
,
2
,...,
ω i Q n
κ α i α i ( α i ) (
=
,
,...,
)
of m
i
1
2
m
.
Let
α σ( j ) = α σ( j ) ,
v
α σ( j ) ),
j
=
1
,
2
,...,
m , then, similar to Theorem 1.45, we
have
Theorem 1.56 (Xia and Xu 2010)
GIFPHAD w 1 2 ,...,α m )
(
1
)
μ α σ( j ) + κ α σ( j ) π α σ( j ) ρ w j
1
ρ
m
1
1
=
,
j
=
1
ρ
κ α σ( j ) π α σ( j ) ρ w j
1
m
1
1
α σ( j ) 1
1
1
v
j
=
1
GIFPHAF w 1 2 ,...,α m )
(
2
)
w j
1
1
ρ
m
μ F n
1
=
,
κ α σ ( j ) α σ ( j ) ( α σ ( j ) )
j
=
1
ρ
ρ w j
1
1
1
m
1
1
v F n
κ α σ ( j ) α σ ( j ) ( α σ ( j ) )
j
=
1
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