EWMA Control Charts Monitoring Normal Variance When No
Standard is Given
Sven Knoth1
Institute of Mathematics and Statistics, Department of Economics and Social Sciences, Helmut
Schmidt University Hamburg, Germany
Most of the literature concerned with the design of control charts relies on perfect knowledge
of the distribution for at least the good (so-called in-control) process. For instance, in order
to monitor the variance of a normally distributed r. v., one usually assumes that the in-control
variance σ02 is known. Given the data is sampled in subgroups of size n, the EWMA control
chart framework is given by:
n
Si2
2
1 X
=
Xij − X̄i ,
n−1
Z0 = z 0 =
j=1
σ02 =
1,
n
1X
X̄i =
Xij .
n
j=1
Zi = (1 − λ)Zi−1 + λSi2 with λ ∈ (0, 1] ,
Lupper = min {i ≥ 1 : Zi > cu } ,
Ltwo = min {i ≥ 1 : Zi > cu or Zi < cl } .
The parameters λ ∈ (0, 1] and cu , cl > 0 are chosen to enable a certain useful detection performance (not too much false alarms and quick detection of changes). The most popular performance measure is the so-called Average Run Length (ARL), that is Eσ2 (L) for the true variance
σ 2 . If the in-control variance, σ02 , has to be estimated by sampling data during a pre-run phase,
then this uncertain estimate effects, of course, the behavior of the applied control chart. Most
of the papers about characterizing the uncertainty impact deal with changed ARL patterns and
possible adjustments. Here, a different way of designing the chart is treated: Setup the chart
through specifying a certain false alarm probability such as Pσ02 (L ≤ 1000) ≤ α, which is quite
common in econometrics literature about structural breaks. Moreover, we illustrate that for setting up EWMA variance charts, it is much more reasonable. Here we describe a feasible way to
determine the control limits in case of unknown parameters for a pre-run series of size N .
1
e-mail: [email protected], web: www.hsu-hh.de/compstat, telephone: +49(0)40 6541 3400