Advances in Risk Management

252 MONITORING COVARIANCES OF ASSET RETURNS

The testing problem for the simultaneous control charts is presented by

H0,t:E(η

(j)

(t))=μη(j) for each j

against

H1,t:E(η

(j)

(t))=μ

(j)

1 =μη(j) for some j.

When the simultaneousT^2 control chart is applied the null hypothesis is
rejected as soon as

max

j=1,...,p

{T

(j)2

t }>h^5 , with T

(j)2

t =(η

(j)

(t)−μη(j))

′#− 1

η(j)(η

(j)

(t)−μη(j))

h 5 determined from the condition that the in-control ARL is equal to a
preselected valueξ.
The simultaneous MC1 scheme is defined similar to the multivariate MC1
control chart. LetS(mj),l=

∑l

i=m+ 1 (η

(j)

(i)−μη(j)) forl,m≥0, and furthermore, let

MC 1

(j)

t =max (||S

(j)

t−nt,t||#η(j)−kn

(j)

t , 0), t≥^1

n
(j)
t is calculated by analogy to (13.11). The MC1 control scheme gives a signal
as soon as the statistic

simMC (^1) t=max
j=1,...,p
MC 1
(j)
t >h^6
exceeds a preselected control limith 6.
ForthesimultaneousMCUSUMweconsiderC
(j)
t =||S
(j)
t− 1 +(η
(j)
(t)−μη(j))||#η(j)
withS
(j)
t− 1 as in (13.12). The control statistic is given by
simMCUSUMt= max
j=1,...,p
MCUSUM
(j)
t (13.18)
where
MCUSUM(tj)=(S(j)
′
t #
− 1
η(j)S
(j)
t )
1
(^2) =max{0,Ct(j)−k}
The simultaneous PPCUSUM control statistic is defined as
simPCUSUMt=max
j=1,...,p
PPCUSUM
(j)
t
with
PPCUSUM
(j)
t =max{0,||S
(j)
t−1,t||#η(j)−k,...,||S
(j)
0,t||#η(j)−tk}