OLHA BODNAR 255Table 13.1Control limits of the multivariate MEWMA, MEWMAas, MC1,
MCUSUM,T^2 charts (section 13.3, in-control, ARL=60)
Type\k= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.3+,r=
MEWMA 213.41 235.84 263.56 494.55 313.70 330.83 344.69 358.55 358.55
MEWMAas 199.55 229.29 261.53 289.25 313.70 330.83 344.40 358.26 358.55
MC1 33.76 31.24 29.16 27.36 25.89 24.50 23.69 22.91 22.16 21.60
MCUSUM 39.62 36.45 33.72 31.47 29.62 28.26 26.74 25.85 24.79 23.97
Table 13.2Control limits of the simultaneous MEWMA, MEWMAas, MC1,
MCUSUM,T^2 charts Section 4 (in-control ARL=60)
Type\k=1.3+,r= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
simMEWMA 12.02 17.32 21.65 25.88 29.21 32.23 34.41 36.07 37.10
simMEWMAas 10.75 16.45 21.33 25.49 29.02 31.91 34.41 36.07 37.10
simMC1 5.41 5.24 5.09 4.95 4.82 4.72 4.56 4.48 4.33 4.22
simMCUSUM 5.60 5.41 5.22 5.07 4.93 4.80 4.66 4.53 4.40 4.28
13.5.2 Behavior in the out-of-control state
For the determination of the out-of-control ARLs we made use of 10^6 inde-
pendent realizations of the underlying process. In Tables 13.3 and 13.4 the
out-of-control ARLs of all control charts within our study are presented. The
corresponding valuesrandkat which the smallest out-of-control ARLs are
attained, are given in brackets. These values should be taken to detect the
specific shift in the mean vector of the process{η(t)}. For a given shift the
ARL of the best chart is printed boldfaced.
Table 13.3 presents the results of the multivariate charts. For the given
choice of the covariance matrix the best result is attained by the MEWMA
control charts. In almost all cases it provides the smallest out-of-controlARL.
On the second to fourth places the multivariate asymptotic MEWMA, MC1
and MCUSUM schemes can be ranked. They are clearly worse than the
nonasymptotic MEWMA approach. The multivariate MEWMAas overper-
forms for the small values of the out-of-control ARL while for the moderate
andlargevaluestheMC1andMCUSUMdesignsshowabetterperformance.
The multivariateT^2 scheme behaves considerably worse. For nearly all shifts
under consideration it has a larger out-of-control ARL than the other charts.
The results of the out-of-control ARLs for the simultaneous charts can
be found in Table 13.4. In all cases the best performance is obtained by the
simultaneous MEWMA control scheme which overperforms the rest of the
control schemes in all cases. It possesses much smaller out-of-control ARLs