210
Appendix D
For
1.
r j < r < r j + 1 ,
G
G
G
+ − −
− −
+
•
+
+
−
=
−
−
−
∫
1
1
1
1
1
r|
1
0
1
EPS
,
1
(
)
(
)(
)
(
) (
)
j
j
r
r
r r
j
j
j
j
j
j
r
r
y u x x
u
u
du
r
r
r r
(D.5)
where
(
) {
}
1
−
+
+


=
+
−


1
1
,
( )
( )
( )
j
j
j
j
j
u
x
H H x
u H x
H x
y
x
(D.6)
For
2.
r < r 1 ,
G
G
G
− −
−
• =
−
−
∫
1
1
1
1
1
|
1
0
1
EPS
1
( )
(
)(
)
(
) ( )
r r
r
r
r
y u x
u
u du
r r r
(D.7)
where
−


=


1
1
1
(
)
( )
y u x
H uH x
(D.8)
For
3.
r > r k ,
G
G
G
G
G
∞
−
•
=
− −
−
+


=
−


− +


− +
×
×
−
=


− +
−


∑
∫
|
1
1
0
1
EPS
1
1
1
1
P
1
(
)
( )
(
)
(
)
(
)(
)
(
)
(
) (
)
k
k
n r
r
r
k
n r
k
r r
n r
k
k
k
n
C
Hx
n r
n r
y u x
u
u
du
N n
n r
r r
(D.9)
where C r is given by Equation D.4 and
( )
( )
( )
{
}
1
−


=
+
−


1
k
k
k
u
H H
u
H x
y
x
x
(D.10)
Theorem 2
Let ranks r 1 < r 2 be given. Let f r 2 (x) denote the conditional density of
X
*
(r 2 )
, given that X
*
(r 2 )
> 0. The probability density function of the ratio of
pool sizes with the specifi ed ranks, for 1 < w < `, is given by
( )
( )
(
) ( )
( ) ( )
( )
( )
( ) ( )
2 1
1
2
2
2
0
2
1
1
1
1
1
r r
r
r
r
r
g w
r r
r
H x H wx
H wx
xh wx f x d x
H x
`
G
G
G
− −
−
−
=
−
−




×
×
∫
(D.11)