Advances in Risk Management

OLHA BODNAR 249

13.3.3 Vector valued CUSUM

Crosier (1988) proposed the multivariate CUSUM control chart, namely
MCUSUM, that is based on the shrinking method. This procedure gener-
alized the univariate proposal of Crosier (1986) to the multivariate situation
by replacing the scalar quantities in the univariate CUSUM recursion by
the vectors in the multivariate case. The idea is to first update the vector
of cumulative sums, then to shrink it towards zero, and, finally, to use the
length of the updated and shrunken CUSUM to test whether or not the
process is out-of-control.
LetCtbe the length of the vectorSt− 1 +(η(t)−μη):

Ct=||St− 1 +(η(t)−μη)||#η

k>0 is the reference value. Then the vector-valued CUSUM scheme for the
vectorη(t)is given by:

St=

{

0 if Ct≤k

(St− 1 +η(t)−μη)(1−Ckt) if Ct>k

(13.12)

fort≥1 withS 0 = 0. The scheme gives an out-of-control signal as soon as
the length of the vectorSt:

MCUSUMt=(St′#−η^1 St)

1

(^2) =max{0,Ct−k}
exceeds a preselected valueh 2 , which is determined with the condition that
the in-controlARLis equal to a fixed valueξ. In practice the last equation has
to be solved by simulations. A practical advantage of the shrinking method
is that the components ofStgive an indication in which direction the mean
has shifted, provided that it is not a false alarm.
13.3.4 Projected pursuit CUSUM
An extension of the CUSUM chart, namely PPCUSUM which is based on
the idea of projection pursuit, was proposed by Ngai and Zhang (2001).
For the directiona 0 with||a 0 || 2 =1 (Euclidean norm), we define the CUSUM
statistic by:
Ca 00 =0, Cat^0 =max{0,Cat−^01 +a 0 ′(η(t)−μη)−k},t≥ 1
Pollak (1985) and Moustakides (2004) showed that the univariate CUSUM
chart possesses certain optimality properties. If the direction of the shift
would be known in the multivariate context then the CUSUM chart based
on the projected observationsa′Xtwould reflect this desirable behavior. The