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

230 Optimizing Optimization

10.4 Section 3: Finite sample properties of estimators

of alpha and tracking error

To compute the biases that we need, it is necessary to calculate various expec-
tations. Consider:

Qi i

a





(,)μμ(,)

β

βγ

Ω^1

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

(10.4)

It is well known that, under normality, the sample mean μμˆ(,)NT1/ Ω,
and μˆ and Ωˆ are independent, where Ωˆ is the Maximum Likelihood esti-
mator of the covariance matrix. First, by Theorem 3.2.11 of Muirhead
(1982) , conditional on μˆ, Qiiˆ^111  (( ˆμμ, )Ωˆ (ˆ, )) has a central Wishart:
WT N 2 (,) 11 /,TQ^1 where Qi i (,)μμˆˆΩ^1 (,).^ The statistic of interest is
given by haˆγγˆ/ˆˆ βˆ^2 and is the first principal element of Qˆ^1. Formally,
we have hQˆ (,^10 )ˆ^1 (,^10 ) and again from Muirhead (1982) Theorem
3.2.5, we have hWTN TQˆ|ˆμ 1 (11 10, (,)/ ^1 (,))^10 and thus, letting
φ 110 10/,TQ(, )^1 (, ) we have

ˆ

|,()

h

TN

φ

φχ^2 ν where ν  1

and consequently, this result holds unconditionally.
Next we examine φ , noting immediately that T φ is the first principal element
in Q

 (^1).
That is,
φ
μμ μ μ



 
1
1
T 11 11
ii
ii ii
Ω
()()()()ˆˆΩΩ ΩΩ ˆ ˆ
(10.5)
Now, μμˆ(,)NT1/ Ω and thus letting ωμ TΩ12/ ,ˆ we have ωN
(,)TIΩ12/μ N. Further, letting c  Ω  1/2 i , we have that:
φ
ωω ω ω
ωωωω






1
1
11
T
c
Tc c
cc cc
Icccc P
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
(())
(10.6)
and it follows immediately that ωωχ PcN (,)^2  1 λ , where λTμ
ΩΩ^12 PThc 12//μ /^ with haγγ β/.
2
Therefore, φχ^1  (,)^2 NTh 1 / and
thus the distribution of hˆ will be given by the following ratio:
ˆ ()
(,)
h
NTh

χ
χ
υ
2
1
2
 /^
(10.7)