OLHA BODNAR 259All these control charts are of the residual type. To construct them we have
made use of the findings presented in Theorems 1 and 2. The multivari-
ate and simultaneous control procedures are independent of the covariance
matrix of asset returns, which constitutes a great advantage of our findings.
No additional information, except the portfolio weights, is required for con-
structing control limits and monitoring the efficiency. Finally, our findings
have financial and statistical significance even for the distribution of the
portfolio asset returns that do not possess the second and higher moments.
The performance of the proposed procedures is obtained within an exten-
sive Monte Carlo study. The best results are reached by the simultaneous
MEWMA control chart, and in second place we can rank the multivariate
MEWMA approach. For that reason we recommend applying either the
multivariate or the simultaneous MEWMA schemes.
APPENDIX
We denoteK′=(μ 1 ,...,μq, 1 ). LetH=(K#−^1 K′)−^1 ={Hij}i,j=1,2andHˆ=(K#ˆ−^1 K′)−^1 =
{Hˆij}i,j=1,2. LetHˆ(−)=Hˆ−^1 ={Hˆ(ij−)}i,j=1,2, whereHˆ( 22 −)= 1 ′#ˆ−^11. ThenwˆM;p=Hˆ( 12 −)/Hˆ( 22 −).
Proof of Theorem 1
From Corollary 3.2.2 of Muirhead (1982) it holds that (n−1)#ˆ∼Wp(n−1,#). Thus,
(n−1)−^1 Hˆ(−)∼Wq−+^11 (n−p+ 2 q+2,K#−^1 K′). Let us denotebˆ=Hˆ 11 (−)−Hˆ( 21 −)Hˆ( 12 −)/Hˆ( 22 −)
andb=H 11 (−)−H( 21 −)H 12 (−)/H( 22 −). Then, from Proposition 1 of Bodnar (2004), it follows that
(n−1)−^1 Hˆ( 12 −)
(n−1)−^1 Hˆ( 22 −)|(n−1)−^1 bˆ∼N(
H( 12 −)
H( 22 −),
(n−1)−^1 bˆ
H( 22 −))Thus
(n−1)−^1 Hˆ( 12 −)
(n−1)−^1 Hˆ( 22 −)−
H( 12 −)
H( 22 −)|(n−1)−^1 bˆ∼N(
0 ,
(n−1)−^1 bˆ
H( 22 −))Hence,Hˆ( 12 −)/Hˆ( 22 −)andHˆ( 22 −)are independently distributed and
√
H( 22 −)√
n− 1 bˆ−(^12)
(
Hˆ( 12 −)
Hˆ( 22 −)
−
H( 12 −)
H( 22 −)
)
|(n−1)−^1 bˆ∼N( 0 ,I)
The righthand side of the last expression does not depend onbˆ. Hence,
√
H( 22 −)
√
n− 1 bˆ−
(^12)
(Hˆ 12 (−)/Hˆ 22 (−)−H( 12 −)/H( 22 −))∼N( 0 ,I) and is independent onHˆ( 22 −). From the other side with
Theorem 3.2.11 of Muirhead (n−1)H( 22 −)/Hˆ 22 (−)∼χ^2 n−p. Combining the results and using
the definition of the multivariatet-distribution the statement of the theorem follows.