COMMUN. STATIST.-SIMULA., 26(1), 107-123 (1997)

PARABOLIC CUSUM CONTROL CHARTS

Stig Johan Wiklund

Department of Statistics, Umel University, 901 87 UMEA, Sweden

Key Words: average run length; shift detection; modified CUSUM charts; CUSUM envelope

ABSTRACT

In this paper some modifications to the traditional "V-mask" CUSUM control

chart, here denoted V-CUSUM, are discussed. In particular, substitution of the

linear decision boundary of the V-CUSUM by a parabolic shaped decision

boundary is studied. The performance, in terms of average run length, ARL. of the

thus defined parabolic CUSUM control chart, P-CUSUM, is explored in a

simulation study. The results for the P-CUSUM are compared with some other

versions of CUSUM charts and with the Shewhart 2-chart. The results show

that the P-CUSUM provides good detection for a wider range of shift sizes than

the other charts under study. Another advantage of the P-CUSUM is its relative

s i i l i c i ty compared to other CUSUM charts, in the respect that the control limits

are given by one parameter only. The P-CUSUM has the disadvantage that it

cannot be handled recurrently and requires the storage of previous observations.

It is argued that the P-CUSUM is better suited to provide estimates of the current

process mean. and may so be preferred to other CUSUM charts in applications

where deviations fiom target are counteracted by adjustments to the process.

C o p y r ~ g l ~ t 1' 1997 by Marce l U e k k e ~ , lnc

108 WIKLUND

The cumulative sum, CUSUM, control chart, due to Page (1954), has during

its four decades of history proven to be a valuable tool in many applications. A

review of some important research papers contributing to the development of the

CUSUM is given by Woodall (1986). A traditional CUSUM control chart can be

adopted by superimposing a so-called V-mask on a plot of cumulated deviations

fiom a target value, and it will therefore be referred to as the V-CUSUM. The

decision rule imposed by the use of a V-CUSUM can be stated in tenns of two

variables, SL, and SH, , cumulating deviations fiom target, and a decision limit,

h, which is chosen in the design of the chart. An alarm is given as soon as

where SL, = max(0 , SL,-I - X, - k) and SH, = max(0 , SH,-, + X, - k ) , with

the initial values typically set to SL, = S H , = 0 . The design parameter k governs

what size of a shifi the chart should be sensitive in detecting. The observations,

X, , are usually the mean of measures of the quality characteristics kom a sample

of items obtained at time t. The variance of X, is here assumed to be known, and

for simplicity taken to be o$ = 1. The presentation shall mainly be confined to

the case when X, are independent and normally distributed random variables,

X, - N ( p , , 1) , and the parameter to be controlled is the mean, p, of the normal

distribution. The target value, po, for the process mean is, without loss of

generality, assumed to be zero, and when p, = po = 0 the process is said to be

in-control. Unless otherwise stated, we consider the two-sided version of the

control schemes, i.e. a deviation from target is to be detected, irrespective of the

direction of the deviation.

To see the correspondence to a V-mask more easily, the decision rule (1) can

equivalently be written in tenns of the cumulative sum of the last t-r observations, t

ST? = X, , and an alarm is given whenever

The linear fimction in the right-hand side of (2) wd on a graphical display look

like a lying "V", which has given the procedure its name. A V-CUSUM is optimal

PARABOLIC CUSUM CONTROL CHARTS : 09

in detecting a shift of a given sue in the mean of a sequence of normal random

variables. The optimality stems from the fact that, in terms of hypothesis testing,

the procedure can be derived from a sequential probability ratio test. However,

the optimality requires the size of the shift to be constant and known, and that the

CUSUM scheme is designed accordingly. These assumptions are in many

applications questionable, and in process control the size of a shift is often

unknown.

The common practice is to design the CUSUM for detection of small to

moderate shift sizes. This habit has lead to the conception that the V-CUSUM is

unsatisfactorily slow in detecting large deviations from the target value. To

overcome this shortcoming, some authors have suggested modifications to the V-

CUSUM Lucas (1982) suggests to combine a Shewhart 2-chart with a V-

CUSUM. The decision rule is then to give an alarm whenever

I x , / > c' or IS^,^ > & + kC(t - r ) for any r;O,l,..,t-1 (3)

where c' is the control limit of the Shewhart part of the decision rule, and the

parameters and kc define the decision rule of the V-CUSUM part (cf. h and k

in (2)). The motivation for this procedure, which we wiU refer to as XICUSUM,

is that the Shewhart part will provide efficient detection of large shifts, whereas

the CUSUM part relatively quickly detects small and moderate shifts. The

combination of the two thus may give good performance for most shift sizes.

On similar grounds Rowlands et al. (1982) propose a "snub-nosed" V-mask,

which in fact reduces to the simultaneous use of two V-CUSUM schemes. Such

schemes were also studied by Dragalin (1995), and following his notation these

schemes are here referred to as 2-CUSUM. An alarm is given as soon as

where h l , k l , h Z , and k2 are the parameters of the two V-CUSUM type

schemes that make up the 2-CUSUM procedure. The idea is that one of the

schemes, i.e. h, and kl , may be designed to detect relatively small shifts, and the

other, i.e. h2 and k 2 , designed to detect larger shifts. Together the schemes may

give relatively good detection for a wide range of shift sizes.

110 WIKLUND

Extending the idea of Lucas (1982), to combine a Shewhart and a V-CUSUM

chart, Bissell(1979) also tries to incorporate the type of complementary decision

rules that are sometimes used in conjunction with the Shewhart chart (cf the

Western Electric rules). Bissell ends up suggesting a scheme corresponding to

using a mask consisting of two sections, a parabolic shaped part providing the

decision boundary for the latest h e time points, and a straight line part giving the

boundary for previous time points. No theoretical motivation was given by Bissell

as to the choice of b c t i o n for the decision boundary. He merely tried several

difFerent masks and chose one that appeared to have the best combination of the

features "appearance, simplicity of construction and ARL performance".

A procedure similar to the one proposed by Bissell had already some years

earlier been suggested by Lucas (1973). Lucas takes the connection of the

CUSUM to a sequential test as a starting point for a discussion, resulting in that

the cumulated sum should not be compared to straight line boundaries, but to

boundaries following a parabolic curve. The possibility of using a parabolic

shaped mask for the CUSUM was also mentioned by Bamard (1959) and

Ewan (1963), but neither one of them develops the idea funher. Lucas is also not

pursuing this idea Il ly, but is suggesting a control chart corresponding to a mask

consisting of two sections; a parabolic shaped nose of the mask and a straight line

tangent to the parabolic part. Tbe decision rule of this procedure, which we shall

denote S-CUSUM, is to give an alarm when

hs + k s ( t - z ) for any r = O,l,..,t - M- 1

S ~ ' > { I C ~ ( * - ~ ) ~ forany r = t - M , . . , t - 1

The parameters hs and ks define the linear (V-CUSUM type) part of the

decision boundary, and K~ defines the width of the parabolic part. It is required

that the parameters are chosen so that the parabolic and linear parts of (5) have

the same height and the same slope in their intersection point, M a . This is 2

obtained if M'= (tcs/2ks) = hs/ks . In the definition (5) we use M = LM'J, i.e.

the integer part of M', so that M is the number of time points for which the

cumulative sum, S,, , should be compared to the parabolic part of the boundary.

The charts of BisseU (1979) and Lucas (1973) can be noted to be similar in

PARABOLIC CUSUM CONTROL CHARTS 1 1 1

structure, altliougl~ arrived upon from somewhat different perspectives. As

mentioned the theoretical arguments of Lucas indicated the use of a purely

parabolic shaped mask. The reason for Lucas to abandon the purely parabolic

shaped mask was that such a scheme would eventually produce an alarm signal

also for a long run of observations whose average is only slightly off target. In

practice, it can be argued that small deviations from target are acceptable and

should not be signalled for by the control chart.

None of the above mentioned procedures have gained any widespread interest,

neither in research nor in applications. A reason may be that the combination of

different mask shapes, i.e. the simultaneous application of different decision rules

for obtaining an alarm, is unnecessarily complicated. For instance, they require the

choice of several control limit parameters. In most applications simplicity is a

virtue, and this may have contributed to rule out the mentioned procedures.

In this paper we will study the properties of a control chart, the decision rule

of which corresponds to applying a purely parabolic shaped mask to the

cumulated sum of observations. This procedure is in the sequel referred to as the

P-CUSUM. In the next section the P-CUSUM is introduced and defined, and a

theoretical motivation is briefly discussed. Properties in tenns of average and

median run lengths are presented, as results f3om.a simulation study. These results

are then used to compare the P-CUSUM with the V-CUSUM, with the modified

CUSUMs defined in (3)-(5), and with the Shewhart 2 -chart.

THE PARABOLIC CUSUM AND ITS RELATION TO OTHER CUSUM SCHEMES

As mentioned in the introduction, the V-CUSUM can, in terms of hypothesis

testing, be viewed as a sequential probability ratio test. Hence, it is optimal for

conducting a test of a simple null hypothesis against a simple alternative. Consider

the sequence of independent random variables, X I , X 2 , . . , X , . Suppose we want

to test the hypothesis that all X I , X 2 , . . , X , have the same distribution, F(Q,) ,

against the alternative that a shift has occurred, so that X I , . . , X, have the

distribution F(QO) and X,,, , . . , X I are distributed according to F(8,) . Lf the

112 WIKLUND

null distribution, F ( $ ~ ) , and the alternative distribution, F ( B ~ ) , are both k n o w ,

one can design a one-sided V-CUSUM scheme that is optimal for detection of

such a change in distribution. A two-sided scheme is obtained by simultaneously

applying two one-sided schemes. For normally distributed variables the

parameters of the V-CUSUM should be chosen as k = 01/2 and h is chosen to

obtain the acceptable type I error of the test. In a process control application, the

parameter B often corresponds to the mean of the outcome fiom a production

process. The null value, 0 0 , is typically presumed to equal the target value for the

process, and is here taken to be zero. The assumptions of the test procedure

hrther imply that at start of production, the process mean equals the target value.

Occasionally the process mean shifts to a known off-target value, Q1, however at

an unknown time-point. After the subsequent detection of the shift the process is

restarted at target value.

However, these assumptions, underlying the optimality properties of the V-

CUSUM, are often unrealistic in process control applications. The process mean

after a shift is often neither known nor constant, in which case the optimality of

the V-CUSUM is not necessarily a major argument, and other control schemes

may be of interest. In the introduction a number of moditications to the V-

CUSUM was mentioned. A theoretical motivation for some of those procedures

was given by Lorden (1971). His results were, however, not referred to by any of

the authors mentioned in the introduction.

The assumption of Lorden is that the distribution after a change point is

unknown, and that one is often interested in detecting any shift larger than some

lower limit O L . In tenns of hypothesis testing, the V-CUSUM corresponds to

performing a test against a simple alternative, F(O,), whereas Lorden's

procedure implicates a composite alternative { F ( B ) ) , 6' z B L . Lorden shows that

using a parabolic shaped boundary for the latest time points, and a straight line

boundary for previous time points, is optimal under a minimax type of criterion

for the time of detection of a shift. For the special case of normally distributed

observations, with target value zero and variance one, the procedure reduces to

using the following boundary for the cumulative sum

PARABOLIC CUSUM CONTROL CHARTS

for ( t - T) 5 M *

for (t - r) > M *

where h* is a parameter controlling the type I error rate of the test. and

M* = 2h * @i2 gives the intersection of the two parts of the boundary. Note from

comparing (5) and (6) that the control scheme of Lorden is equivalent to the S-

CUSUM suggested by Lucas (1973). Lorden, however, makes the connection to

hypothesis testing explicit by defining the decision boundary as a h c t i o n of the

hypothesis value, e L . Although theoretically appealing, this semi-parabolic

CUSUM d e r s from the disadvantage of being somewhat complicated, requiring

the choice of both h* and Q L , the calculation of the intersection point, M*. and

the handling of a control boundary with Werent shapes.

The semi-parabolic CUSUM of Lorden is appreciably simpaed if the

composite alternative hypothesis is taken to be {~(o)), B > Q0 (instead of

B > BL ). This could be viewed as a limiting case of the semi-parabolic CUSUM

as Q L -+ Q O . The intersection point, M*, is thereby moved backwards, resulting

in a purely parabolic shaped boundary. The decision rule is then simplified, and

introducing the notation K for the design parameter of this procedure, an alarm

signal is given whenever

s ( t - ) for any r=0,1,..,t-l (7)

presuming that two symmetric one-sided schemes are applied to obtain a two-

sided scheme. A control scheme as defined in (7) wiU be referred to as a parabolic

CUSUM, denoted P-CUSUM. It is clear &om comparing (5), (6) and (7) that an

advantage of the P-CUSUM is its simplicity in terms of design. Only one

parameter, K, has to be chosen to determine the control limits. This value \dl

typically be taken to obtain an acceptable rate of false alarms, i.e. to obtain a

required in-control average run length.

After the control chart has indicated that a shift has occurred, it would often

be of interest also to know when the shift occurred, and to get an indication of the

114 WIKLUND

magnitude of the shift. Under the same assumptions as used in motivating the P-

CUSUM, i.e. that the process mean is unknown after a shift. the maximum

likelihood estimator. b, of the time of a shift is given by the value of r

maximising ( t - T ) X ; , , where X,, = ( t - r ) - ' s I , (e.g. Hinkley ( 1970) and

( 197 1)) Equivalently, i, is the value of r maximising

Comparing (7) and (8): we see that the observation falling outside the mask of a

parabolic CUSUM is in fact an ML-estimator of the time of a shift. A

corresponding estimator of the process mean after the shift is given by the

average of the observations from the time G up to the time of the alarm

Tlie fact that the IWUSUM provides a reasonable estimator for the current

process mean hrther motivates a study of the P-CUSUM as a control chart

method. The preceding discussion in particular indicates that the P-CUSUM may

prove u s e l l in situations where a deviation fiom the target value is counteracted

by an adjustment. In such situations, the current process mean is typically not

known with certainty, neither before nor after the alarm, and an appropriate

estimator of the current process mean is vital. Such an estimator is, due to the

difference in assumptions underlying the procedures, more likely to be provided

by the P-CUSUM than by the ordinary V-CUSUM.

SIMULATION RESULTS AND ARL COMPARISONS

The properties of the P-CUSUM in terms of average run length, ARL, have

been studied in a simulation study. The observations, X,, were taken to be

independent and normally distributed, with target value zero and unit variance.

The process was assumed to be initially out of control, i.e. XI, X 2 , ...- N(6 ,1 ) ,

where 6 is the shift size. Evaluations are made for values of the control limit

parameter, K, ranging from 2.8 to 4.0, and for the shift size, 6, ffom 0 to 4.0. For

PARABOLIC CUSUM CONTROL CHARTS 115

TABLE I. Estimated average run length, ARL, for the P-CUSUM control chart.

(The results are based on slmuiat~on of 10000 shift detections for each conlbinat~on of control limit and shift slze. The standard deviation for the ARL estimates ranges from 0.3% of the ARL In the upper right corner of the table, to 1.1% in the lower left corner.)

each combination of and 8, 10000 shift detections were simulated. Results from

the simulations are shown in Table I.

As an aid in designing a P-CUSUM chart, and to facilitate comparisons with

other control charts, we have calculated the values of K giving the in-control

average run length, ARh, for some multiples of hundred. The values were

obtained through interpolation in Table I. The interpolation was camed out by

fitting a quadratic fimction to the logarithm of the entries of the first column, and

the results are shown in Table 11.

We can note from Tables I and I1 that a choice of K 111 the interval 3.4-3.5

corresponds to the ARLO for some frequently applied control charts. The ARLO

for the 3o-Shewhart chart is obtained by the P-CUSUM for K = 3.4. and tbe

ARLO of the popular V-CUSUM with h=5 and k 0 . 5 is obtained by ,Y - 3.48. In

a subsequent comparison between some control schemes we use K = 3.5, to

have ARLO - 500.

116 WIKLUND

TABLE 11. Control limits, K, of the P-CUSUM control chart corresponding to some values of the in-control average run length, ARLO.

TABLE III. Control schemes used in the ARL comparisons

control scheme

P - c u s m

v-CUSUM: 1

v-CUSUM:2

Shewhart X - chart

X / c u s m

2-CUSUM: 1

2-cusuM:2

S-CUSUM

parameter values method of obtaining ARL

Simulation *

Solving integral equations

Solving integral equations

Evaluating the normal distribution fimction

Markov chain

Markov chain

Markov chain

Simulation *

source for ARL

Lucas (1982) ***

Dragalin (1995)

Dragalin (1995)

* The results for P-CUSUM and S-CUSUM are based on s~mulation of 10000 shift detections

** Note that the Shewhart ,y -chart is equivalent to a V-CUSUM with h=O, k=3.09.

*** Values are obta~ned by interpolating between the ARL Dven by Lucas (1982) for hc =5.0 and h 4 . 1 , respectively. **** M1=11.9, i.e. parabolic boundary for the last 11 time points and linear boundary for previous time points.

PARABOLIC CUSUM CONTROL, CHARTS 117

The results for the P-CUSUM reported from the silnulation study have been

compared to the Shewlial-t X-chart, the V-CUSUM and the modified versions

of the CUSUM charts, defined in (3), (4) and ( 5 ) . A summary of the compared

control schemes is given in Table 111, containing for each coutrol scheme, the

parameter values, how the average run length was calculated and, when

appropriate, also from where the ARL was taken. Note that two designs are

chosen for the V-CUSIJM and 2-CUSUM, respectively. and in each case the

two designs are denoted by a trailing ": 1 " and ":2" on the name.

For each of the control charts, parameter values are chosen to approximately

give an ARLo=500 The ARL for each control chart, and for different shift sizes.

are shown in Table IV In this table all the studied control schemes can also be

compared with the so-called CUSUM envelope, (Rowlauds et al. (1982),

Dragalin (1994)). For a given ARLO, the CUSUM envelope is defined as the

average run length, A R L s , for a V-CUSUM designed to detect a shift of size S and guaranteeing the gibe0 ARLO Hence. the envelope to be considered here

can be defined as

Since the V-CUSUM is optimal for a given 8, the CUSUM envelope gives the

lowest obtainable A R L for each shift size. It can therefore serve as a benchmark

for evaluating control charts.

To facilitate the comparison and evaluation of the dserent control schemes,

we do in Table V, for each control scheme and shift size, report by how many

percent the ARC exceeds the CUSUM envelope benchmark. For each control

scheme the values are, in both Tables IV and V, given on light shaded

background for those shift sizes where the ARL is more than 10% higher than

the envelope. Values are on heavier shaded background when the ARL. is more

than 20% higher than the envelope. Hence, the range of 6 for which the values

are given on a white background can be viewed as a "window" of shift sizes in

which the control chart provides good shift detection.

118 WIKLUND

TABLE IV. Average run length, ARL, for each of the studied control schemes. ,

H . 2 5 0.5 0.75 1.0 1.5 2.0 2.5 3.0 3.5 4.0

P-CUSUM 214,, 369 18-5 1 1 4 5 77 3 63 2 56 1 96 1 56 1 32

TABLE V. The excess (in %) in ARL for each control scheme, compared to the lower bound given by the CUSUM envelope.

l H . 2 5 0.5 0.75 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1

CUSUM envelope

(A cell 111 the table is light shaded if .ML, > 1 1 * A R ~ ( G ) , and heawer shaded if

:tKL, > 1.2 * A R L * ( ~ ) , where ARI,; is the value for the CUSUM envelope.)

83 31.1 16.6 10.5 5.45 3.41 2.39 1.79 1.42 1.20

( A cell ~n the table is light shaded if ARLs > 1.1 * . 4 ~ ~ * ( 6 ) , and heav~er shaded if

.-IRL, > 1.2 * .4~1,*(6) , where ARI,:, is the value for the CUSUM envelope )

PARABOLIC C U S U M CONTROL CHARTS 119

Il le P-CUSUM is fi-om Table V seen to have, among the coutl-01 schemes

under study, the widest range of shift sizes for wluch it provides good detection.

For any shift size, 6 > 1.0, the excess of the P-CUSUM ARL is less than 10% of

the lower bound given by the CUSUM euvelope. The P-CUSIJM should

therefore in particular be useful when only little is known about anticipated shift

sizes. Moreover, it is clear that in applications where one presumably has a good

idea of the size of shifts, the loss from applying a P-CUSUM is vesy small.

The V-CUSUM: I is designed for detection of shifts 6=1 .O, and V-CUSUM:2

for detection of S 2 . 0 . For exactly those shift sizes, the V-CUSUM schemes

coincide with the envelope benchmark, thus giviug a somewhat lower ARL than

the other schemes. However, this has the price of a relatively bad performance

for other shift sizes. The window of good performance, as illustrated in Tables

IV and V, is seen to be rather narrow for the V-CUSUMs. For example, the

popular V-CUSUM: 1, with h=5.0 and k=0.5, gives a somewliat lower ARL than

tlie 1'-CUSUM only for t11e shift sizes 6=0.75 and S=1.0. To assume that the size

of any slliA is known in advance with such a precision is indeed restrictive.

The Shewhalt 2-chart is well-known to be inefficient for detecting small

and moderate shifts, and this is also clearly illustrated in Tables IV and V. The

A R L is less than 20% over tlie envelope benchmark only for 6 1 3 . 5 , and it is

only for these very large shifts that the Shewhart chart compares favourably with

the P-CUSUM. However, the lnain advantage of the Shewhart chart is its

outstanding simplicity, and iu many applications this clearly outweighs the

disadvantage of the unfavourable ARL properties.

The XICUSUM scheme in this study ~ n a d y mimics the V-CUSUM. I , the

scheme to which the Shewhart decision rule was actually added to obtain the

combiued scheme Only for really large shifts, 6 2 3 0 , the XICUSUM reduces

the ARL of the V-CUSUM I It can be questioned if this minor unprovement is

worth the price of the increased complexity by the simultaneous use of two

decision rules.

A similar concern holds true for the 2-CUSUM schemes uuder study. The

range of delta for which the 2-CUSUMs provide rather good ARL values is

120 WIKLUND

certainly somewhat wider than for a corresponding V-CUSUM, but it is in an

application not at all obvious that this improvement is worth the extra

complexity of the 2-CUSUM. Moreover, the 2-CUSUMs do not provide the

uniformly good performance over such a wide range of S as do the P-CUSUM.

The performance of the semi-parabolic, S-CUSUM, is similar to the P- CUSUM. This is to be expected since the control boundaries for the two

schemes are of the same shape for the latest time points. The difFerence lies in

how the schemes treat the past history of observations. The purely parabolic

shaped boundary of the P-CUSUM thereby results in a lower ARL for very small

shift sizes. The similarity in ARL properties imply that the choice between the S-

CUSUM and the P-CUSUM ought to be based on other considerations. One

important argument may then be the relative simplicity of the P-CUSUM,

requiring the choice of only one control limit parameter, K.

DISCUSSION

The V-CUSUM is optimal for detecting a shift of a certain size, and it could

be argued that there is no need to consider any other CUSUM scheme. In

practice, however, the size of a shift is typically neither known nor constant, so

the optimality property of the V-CUSUM is not necessarily a major argument. It

is seen from Tables IV and V that the V-CUSUM as expected has a low ARL for

shifts of about the size that it was designed for detecting. For shills sizes

substantially different from this value, the V-CUSUM is clearly outperformed by

some of the other schemes, and in particular by the S-CUSUM and P-CUSUM.

Hence, the latter can in some sense be viewed as providing a minimax type of

control scheme, particularly suitable when little is known about anticipated

deviations from target.

An important aspect of any control scheme is its s i i l ic i ty , and thereby its

applicability to practical problems. Here is an obvious advantage of the P-

CUSUM that it only requires the choice of one parameter, K, whlch is directly

associated with the in-control ARL. Hence, the practitioner only needs to decide

what rate of "false alarms" he is willtng to accept. The design of the control

scheme, in terms of control lirmts, is then given.

PARABOLIC CUSUM CONTROL CHARTS 12 1

A disadvantage of the P-CUSUM is that it cannot be handled recurrently (as

can the V-CUSUM through the definition (1)) and in principle requires all

observations since the previous alarm to be stored. However, as of today most

CUSUM applications are computerised and even the most modest personal

computer has the capacity of storing and handling thousands of observations. In

an application, the production will now and then be reset, and the control chart

will be restricted to only those observations resulting fiom the current

production setting. Consequently, the number of observations that have to be

handled by the P-CUSUM will normally never even come close to the limit given

by the computational system. Moreover, the production runs in industry tend to

become shorter, which fkrther decreases the negative effect of the "non-

recurrent" property of the P-CUSUM. If the storage of many observations is

anyhow a problem, one solution may be to truncate the decision fuoction and

only consider the last N observations. Experience based on simulation

experiments indicates that the change in ARL properties of the P-CUSUM by

such a truncation is in most cases marginal also for rather small N. The effect of

truncation is appreciable only for small shifts and mainly for high values of K

Hence, the combination K-4.0 and &O 25 is the case in Table I, which will be

influenced most by a truncation of the decision limit. Even in this case a

truncation to N=200 observations will increase the ARL less than 1%. ']The effect

on the ARL of a truncation of the decision limit is in fact much smaller for larger

6 or smaller K.

The P-CUSUM provides rather good detection for almost any size of a shift.

This can sometimes be regarded as an undesired property, since it will give alarm

signals also for deviations fiom the target value of negligible size However, any

control chart will occasionally give an alarm signal also for very small shifts. An

important point is that such an alarm from a P-CUSUM is likely to be based on a

rather long run of observations, whose average is close to target \due , hence

giving an indication that the slufl is not an important one This property is of

course related to the fact, given in (8) and (9), that an ML-estimator of the

current process mean can be obtained from applying a P-CUSUM An alarm

from a V-CUSUM, however, is most often triggered by a rather short nu] of

122 WIKLUND

large observations, hence not yielding the same possibility to distinguish between

alarms caused by large and small shifts.

Control charts are often used, not only for the detection of an out-of-control

situation, but also for providing a basis for conducting counteracting adjustments

to the process. In these situations it is reasonable to assume that the state of the

process is always unknown, also immediately after an alarm/adjustment. This is

because the amount of adjustment needed to bring the process to the target

value cannot be known exactly, but have to be estimated from the sampled

items. The process will then always, more or less, deviate fi-om target, and an

alarm signal may be triggered, and also desired, not only when a shift has

occurred to the process, but also when the previous adjustment was

unsuccessful. In such applications the possibility of a control chart to indicate the

amount of deviation !?om target is of great importance. Hence, the fact that the

P-CUSUM is directly related to an estimator of the process mean can be

expected to be a valuable property. The effect of using different CUSUM

schemes in t h s manner, for controlling a process through adjustments at times of

alarms, would be an interesting topic for further research

ACKNOWLEDGEMENTS

This research was camed out during a visit at the Imtitut fur Angewandte

Mathematik und Statistik, University of Wiirzburg, Germany. Financial support P

for the visit from The Swedish Institute, Stockholm, Sweden, and fiom the

Faculty of Social Sciences, University of Umei, Sweden, is gratefully

acknowledged. Thanks are also due to Vladimir Dragalin for calculating some of

the ARL values reported.

BIBLIOGRAPHY

Barnard, G.A. ( 1 959). "Control Charts and Stochastic Processes," Journal of the Royal Statistical Sociep, Ser. B, Vol. 21, 239-257.

Bissell, A.F. (1979). "A Semi-Parabolic Mask for CUSUM Charts," The Statistician, Vol. 30, 1-7.

PARABOLIC CUSUM CONTROL CHARTS 123

Dragalin. V. ( 1994). "Optimal CUSUM Envelope for Monitoring the Mean of Nonnal Distribution," Ecorzonzrc Qlrallty Corltrol, Vol. 9, 185-202.

Dragalin, V. (1995). 'The Design and Analysis of 2-CUSUM Procedure," Research Report of the Wiirzburg Research Group on Quality Control, No. 54.

Ewan, W.D. ( 1 963). 'When and How to Use Cu-Sum Charts," Techt~on~etrrcs, Vol. 5 , 1-22.

Hinkley, D.V. (1 970). "Inference About the Change-Point 111 a Sequence of Random Variables," Biometrik, Vol. 57, 1- 17.

Hinkley, D.V. ( 197 1). "Inference About the Change-Point From Cumulative Sum Tests," Riometrrka, Vol. 58, 509-523.

Lorden, G (197 I). "Procedures for Reacting to a Change in Distlibution." The Atn~als ofMathenzatrca1 Slatrstrcs, Vol. 42, 1897- 1908

Lucas, J.M. ( 1973). ''A Modified V-Mask Control Scheme," Tec/it~otriett~~cs. Vol. 15. 833-847.

Lucas, J.M. (1982). "Combined Shewhart-CUSUM Quality Control Schemes," Jourtlul ofQuali0~ Technology, Vol. 14, 5 1-59.

Page, E. S. ( 1954). "Continuous Inspection Schemes," Bron~etrrh, Vol. 4 1 . 100-1 15.

Rowlands, R. J., Nix, A.B.J., Abdollahian, M.A. and Kemp, K. W. ( 1982). "Snub-nosed V-mask Control Schemes," The Statisticlan, Vol. 3 I , 133- 142.

Woodall, W.H. (1986). ''The Design of CUSUM Quality Control Chalts," Jozirtzal of Qualrty Technologv. Vol. 18, 99- 102.

R e c e i v e d Nove~nber , 1995; R e v i s e d A p r i 1 , 1996