Some properties of averaging simulated optimization methods 231
where the two χ 2 variables in Equation (10.7) are independent. This result also
appears in Stein (2002). Thus, pdf()hˆ can be easily found using results related
to noncentral F distributions. In this regard, we have from Johnson and Kotz
(1972, p. 191) that the pdf ((1/ h ) g ) is, noting B( ) and 1 F 1 ( ) to be Beta and
confluent hypergeometric functions respectively,
pdf g
eg
BgFTh NNT
NN
() N
,( ) ,,
/2 1
1
221121
21
11
21
ν 22
νν
() hhg
()1g⎛
⎝⎜⎜
⎜⎜⎞
⎠⎟⎟
⎟⎟Using this result and simple transformations, one can readily derive the pdf
density functions for the quantities of interest, viz, αλˆ(1/ hˆ). Tracking
error TE ()1/λ hˆ and the Information Ratio IR ()1/ hˆ Thus, we have:.
pdf TE y
ye y
ByTh
NN
() N
()
,( )
2
122221
1
2222 121
21λλ
ν λ
ν/()FFNNT
hy
y122
221
21(^221)
νλ
λ
,,
()
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
, y 0
From these pdfs or via that of hˆ or g, we can easily find moments. That is,
since
ˆ
() [( )][ )]
(,)()
(,) ()
hg
Eg E E
NTh
k
NTh
kk
1
1
22
1
22
χ χ
χχ
ν
ν
/
/
/
and since
E
eN
kF
N
NTh
k
Thk
[(χ(,) ) ] N
(^21)
2
1
2
11
21
2
1
2
/
/
Γ
Γ
()
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟ kk
N T
,;h
1
2 2
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
pdf
e
B
F
N
Th
N
N
() N
()
,( )
αω
λλω
λω
ν
ν ν
ˆ
/2 1
1
22
11
21
21
1
1
2
()
,,,
()
,
NT
h
1
221
0
λω
λω
ω
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
pdf
e
B
F
N
Th
N
N
() N
()
,( )
αω
λλω
λω
ν
ν ν
ˆ
/2 1
1
22
11
21
21
1
1
2
()
,,,
()
,
NT
h
1
221
0
λω
λω
ω
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
pdf IR x
xe x
Bx
F
N
Th
N
N
() N
()
,( )
2
1
1
221
1
22
2 11
21
21
/
()ν ν
νν
2
1
(^221)
0
2
,,() 2 ,
NT
h
x
x
x
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
pdf IR x
xe x
Bx
F
N
Th
N
N
() N
()
,( )
2
1
1
221
1
22
2 11
21
21
/
()ν ν
νν
2
1
(^221)
0
2
,,() 2 ,
NT
h
x
x
x
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟