Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Some properties of averaging simulated optimization methods 233

For other quantiles, we have:

ETE F

N

Th

N

() N () ,;













ΓΓ

ΓΓ

1

2

1

22

1

2

1

2

11

1

2 2

1

2

2

()()

()

⎛

⎝

⎜

ν

λν/

⎜⎜⎜ /

⎞

⎠

⎟⎟

⎟

and

EIR()λETE( )

Also note that since EETEE() ( )σαλ ()

(^2) ^21 ˆ,
an unbiased estimator of σ^2 is
easily derived to be:
ˆ ˆ
()
σ
λ
ν
α
λ
(^2) ^12   ^1
T
N
T
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
While little progress can be made with exact expressions for the expectation of
TE and IR, we can get more insight by considering approximations. Jobson
(1991) gives similar results in that he derives the means and variances of a, β ,
and γ and h ^1 and determines their marginal distributions. Stein (2002) also
derives some formulae similar to ours.
We now examine a situation in which both N , the number of stocks or
assets, and T , the sample size, increase in such a way that the ratio ( N  1)/ T
remains constant. We note that α , β , and γ and hence h also depend upon N
and thus asymptotics here require that the terms limit to a constant, or at least
we need to make assumptions about 1/ h as a function of N. For the moment,
we shall not consider this possible influence. Thus, we now let ( N  1/ T)  n
so that N  1  T. n. By letting T → , we can readily see the effect on the
moments of large N and T. For αˆ, we have from our exact result:
E
Tn
Tn hn
()
(( ) )
α
λ
ˆ 


12
1
⎡ 1
⎣
⎢
⎢
⎤
⎦
⎥
⎥
and therefore as T → , we find:
E
n
nh n
()
()
α
λ
ˆ →
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1
1
1
 1