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OTE/SPH
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JWBK119-25
August 31, 2006
388 CUSUM and Backward CUSUM for Autocorrelated Observations
Note that in the above equation, BCUSUM j+1 = 0 for j + 1 > n. From (25.13) it is
evident that CUSUM j and BCUSUM j+1 are symmetric and the point of symmetry
is at CUSUM n /2. By the virtue of their symmetry, one can therefore implement the
CUSUM mask for the BCUSUM scheme and vice versa. In the following section, we
will derive a BCUSUM mask based on a UMP test.

25.3.1 Parabolic mask for CUSUM
√
The BCUSUM limit for the ( j + 1)th observation is z * n − j. By symmetry, we can
therefore derive the following CUSUM scheme for detecting an increase in the mean:

*

CUSUM j ≤−z σ n − j + CUSUM n . (25.14)
For a two-side scheme one may implement:
√
CUSUM j ≥ z * σ n − j + CUSUM n ,
√ (25.15)
CUSUM j ≤−z * σ n − j + CUSUM n .
Similar to the BCUSUM scheme, the value of z * in (25.14) is chosen such that the
CUSUM achieves a pre-specified in-control ARL.
In using (25.14), we assume that CUSUM 0 = 0. With this assumption, the origin is
also used to decide an out-of-control situation (i.e. it must also lie within the CUSUM
mask when the process is in control). For a V-mask, for example, if the first observation
is greater than h + k, the origin is the point that will generate the signal. Similarly, for
the parabolic mask of (25.14), the first observation will signal a change when the y 1 is
greater than z * .

25.3.2 Numerical illustration
To throw more light on the relationship between the CUSUM and BCUSUM, we
illustrate the results in the preceding section using a numerical example. The data
in Table 25.2 were obtained from Woodall and Adams. 23 In the table, the first 10
observations are random variables from N(0,1). The next five observations are from
N(1,1). The CUSUM is calculated using the conventional method while BCUSUM
values are calculated using equation (25.8); the corresponding masks are computed
using equations (25.4) and (25.8), respectively.
Visually analyzing the data in Table 25.2, one cannot easily detect the symmetry of
the CUSUM and BCUSUM. Their relationship becomes obvious once plotted against
time (see Figure 25.2). From Figure 25.2, one can easily appreciate the symmetry of the
two CUSUM schemes which is used to establish the parabolic mask for the CUSUM.

25.3.3 ARL performance of CUSUM when the BCUSUM mask is used
15
Atienza et al. compare the ARL performance of a CUSUM parabolic mask with other
CUSUM mask schemes. By using Monte Carlo simulation with 5000 iterations, they
give the ARL and standard deviation of ARL (SARL) of the various CUSUM schemes
for selected σ shifts in mean. The sequence of normal random variables was generated
24
using the IMSL Statistical Library. The comparisons are replicated in Table 25.3.