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346 Simultaneous Monitoring of the Mean, Variance and Autocorrelation
while the autocorrelation function is given by

1, k = 1,
ρ k =
φ 1 ρ k−1 , k ≥ 2.
The autocovariance and autocorrelation function of the general ARMA(p, q) process
14
can be found on p. 76 of Box et al. . The procedures for estimating the autocovariance
function are discussed in detail by Brockwell and Davis 13 and Box et al. 14
From the above characteristic of the Gaussian ARMA process, we know that
−1

H n = x x (22.1)
0
2
has a χ distribution with n degrees of freedom. This statistic is sensitive to changes
in both the mean vector and the autocovariance matrix. If we divide {x 1 , x 2 , x 3 ,...}
into disjoint series of measurements of length n ({x 1 = (x 1 , x 2 ,..., x n ), x 2 = (x n+1 ,

x n+1 ,..., x 2n ),...}), we can choose n large enough such that the correlation among

2
x 1 , x 2 ,... becomes negligible. Hence, we can implement the typical χ control

chart for monitoring an ARMA process. Since this control chart is analogous to the
2
multivariate T control chart for monitoring products or processes with several cor-
related quality characteristics, 10,15 one may extend the application to the multivariate
CUSUM or exponentially weighted moving average (EWMA) for monitoring auto-
correlated processes. 16−19
2
In implementing the above χ control chart, after calculating an H statistic, one
needs to wait for all the following nobservations before computing the next H statistic.
This can slow down the detection of process changes, especially when n is large. One
alternative is to calculate the H statistic for the latest n observations, that is, using the
following sequence of vector of measurements:
*
*
*

x = (x 1 , x 2 ,..., x n ) , x = (x 2 , x 3 ,..., x n+1 ) , x = (x 3 , x 4 ,..., x n+2 ) ,....
1 2 3
In what follows, we will call the H chart based on the above sequence of measurements
the moving H chart or MH chart. For this case, the resulting H statistics are no longer
independent. Thus we cannot establish the control limits using an approach similar
2
to the construction of the T control chart. We can use simulation to determine the
control limits that will achieve a desired in-control ARL.
22.3 ARL PERFORMANCE
The monitoring strategy described in the previous section is obviously a Shewhart
scheme. When n = 1, one can easily verify that the proposed control charting strategy
becomes a special case of a Shewhart control chart with modified control limits. Thus,
in the ensuing sections we will only compare the ARL performance of the proposed
scheme with similar Shewhart monitoring schemes (SCC and its rival SACC).

22.3.1 Sensitivity in detecting a mean shift
The sensitivity of the MH chart is dependent on the size of the moving window n.
Using Monte Carlo simulation, we analyze the effect of n on the ability of the MH