248 MONITORING COVARIANCES OF ASSET RETURNS13.3.1 T^2 Control chart
The multivariate Shewhart control chart (Hotelling, 1947) is based on mea-
sure the distance between the vector of observation and the target mean of
the process. Becauseμηis known (see (13.7)), the control statistic is given by:
Tt^2 =(η(t)−μη)′#−η^1 (η(t)−μη)The null hypothesis of no changes should be rejected at timetifTt^2 >h 1 ,
which is determined under the condition that the in-control ARL is equal to
a preselected valueξ. For further discussion of the properties of theT^2 chart
we refer to Alt and Smith (1988) and Runger, Alt and Montgomery (1996).
13.3.2 MC1
The most frequent application of CUSUM schemes is the case when a detec-
tion of small changes in the process parameter is of interest. An important
property of the CUSUM chart for univariate data is that it can be derived
via the sequential probability ratio test of Wald (1947). In the multivariate
case it leads to a control scheme which depends on the direction of the shift
(Healy, 1987). Because this is a very unpleasant property, three other types
of CUSUM charts for multivariate observations have been introduced by
Crosier (1988) and Pignatiello and Runger (1990), Ngai and Zhang (2001).
A more detailed discussion of these schemes can be found in Bodnar and
Schmid (2006).
LetSm,l=
∑l
i=m+ 1 (η(i)−μη) forl,m≥0. For an arbitrary vectorxand a
positive definite matrixCwe define the norm||x||C=
√
x′C−^1 x.
The MC1 control chart is constructed by applying the MC1 control scheme
of Pignatiello and Runger (1990) to the processη(t):
MC (^1) t=max (||St−nt,t||#η−knt, 0), t≥ 1
wherentbeing the number of observations since the most recent reconstruc-
tion of the CUSUM chart:
nt=
{
nt− 1 + 1 if MC (^1) t− 1 > 0
1 if MC (^1) t− 1 = 0
(13.11)
fort≥1 withMC 10 =0.
WhenMC (^1) texceeds an upper control limit then the process is considered
to be off-target. For the interpretation of the out-of-control signal one can
consider the components of theSt−nt,t/nt+μηwhich indicate the direction
of the shift provided that it is not a false alarm.