Some properties of averaging simulated optimization methods 239
where
QR RˆK (ˆ,μμ)Ωˆ^1 (ˆ, )with
QWTNKˆ (,)
T
KK^1 1 QK^11
Following earlier results, we now define:hQˆ (,)ˆ (,)
KK
10 ^110and we have immediately, corresponding to Equation (10.7):
ˆ ()
(,(/))hK TNK
NKThKχ
χ2
2
where hQKK(, )’^1 (, )10 with Q (^10) K ( μ ,R ) Ω ^1 ( μ ,R ).
Thus , by a simple substitution into our earlier results, we can readily spec-
ify the exact distribution and moments of αˆ,IR, and TE. That is, we merely
replace N 1 by N K and h by h K. As intuition suggests, increasing the
number of restrictions is exactly the same as reducing the number of assets.
However, the noncentrality parameter h K will change as the constraints change.
This is clear from the following. If we let:
R
r
r
r
KN
K
1
2
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟ where r 1 is a 1 N vector.
Then,
Q
R
RRR
a
K
K
μμμ
μ
ββ β
β
ΩΩ
Ω
11
11
11
Ω
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
γγγ γ
βγ γ
β
β
(^1 121)
K
KK KK
a
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
ΓΓ
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
and h K will be the (1,1)-th element of the inverse of the ( K 1)-th principal
minor of Q K.
That is,
haK
()ββΓ^11