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392 CUSUM and Backward CUSUM for Autocorrelated Observations
mean; and for the same reason as in i.i.d. case, z α in (25.18) is not used as the α-value
cannot be interpreted as the probability of eventually obtaining a false alarm. 27
Note that (25.19) can also be written as

n
|h|
(n − j + 1) 1 − γ (h), j = 1, 2,..., n.
y i > z α
n − j + 1
i= j |h|<n− j+1
The left-hand side is just the sum of the n − j + 1 most recent observations, which is
the (BCUSUM) shown in the previous section. We can therefore test the presence of a
step shift in mean by applying the following scheme:
√
n
BCUSUM ≥ z * υ n− j+1 n − j + 1, j = 1, 2,..., n, (25.20)
j
where

|h|
2
υ n− j+1 = 1 − γ (h).
|h|<n− j+1 n − j + 1
Similar to (25.9), to detect both an increase and a decrease in mean, the following
scheme may be implemented:
√
n
BCUSUM ≥ z * υ n− j+1 n − j + 1,
j
√ j = 1, 2,..., n. (25.21)
n
BCUSUM ≤−z * υ n− j+1 n − j + 1,
j
Graphically, the control limits for the BCUSUM in (25.20) and (25.21) resemble a
parabolic mask. Note that in implementing the above BCUSUM schemes, one only
needs an estimate of the autocovariance function of the series being monitored.
25.4.2 Mask representation
Implementing the BCUSUM from either equation (25.20) or (25.21) for the on-line
detection of mean shift implies that we need to calculate n BCUSUMs at every time
period t. In addition, note that n increases as more observations become available.
The BCUSUM procedure is therefore computationally intensive when implemented
on-line. Interestingly, we can express the BCUSUM decision rules shown in equations
(25.20) and (25.21) using the typical CUSUM representation, which has been shown
Section 25.3 (see the derivations of equation (25.13)) to be as follows:
*
n

n
CUSUM ≤−z υ n− j n − j + CUSUM , j = 1, 2,..., n, (25.22)
j n
and
√
n n
CUSUM ≥ z * υ n− j n − j + CUSUM ,
j n
√ j = 1, 2,..., n, (25.23)
n n
CUSUM ≤−z * υ n− j n − j + CUSUM ,
j n
respectively. Similar to the BCUSUM scheme, the z * in equation (25.22) or (25.23)
is chosen such that the CUSUM chart achieves a pre-specified in-control ARL. The
parabolic CUSUM scheme for monitoring a sequence of i.i.d. normal random variables
shown in Section 25.3 is a special case of equation (25.23).
0
In using equation (25.22) or (25.23), we also assume that CUSUM = 0 as in the i.i.d.
0
case. Similarly, for the parabolic masks equation (25.22) or (25.23), the first observation