International Journal of Pure and Applied Mathematics Volume 37 No , IMPROVED DATA DRIVEN CONTROL CHARTS

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1 International Journal of Pure and Applied Matheatics Volue 37 No , IMPROVED DATA DRIVEN CONTROL CHARTS Wille Albers 1, Wilbert C.M. Kallenberg 2 1,2 Departent of Applied Matheatics Faculty of Electrical Engineering, Matheatics and Coputer Science University of Twente P.O. Box 217, AE Enschede, 7500, THE NETHERLANDS 1 e-ail: w.albers@ath.utwente.nl 2 e-ail: w.c..kallenberg@ath.utwente.nl Abstract: Classical control charts for onitoring the ean are based on the assuption of norality. When norality fails, these control charts are no longer valid and serious errors often arise. Data driven control charts, which choose between the noral chart, a paraetric one and a nonparaetric chart, have recently been proposed to solve the proble. They also correct for estiation errors due to estiation of the paraeters involved or, in the nonparaetric chart, for estiation of the appropriate quantiles of the distribution. In any cases these data driven control charts are perforing very well. However, when the data point towards the nonparaetric chart no satisfactory solution is obtained unless the nuber of Phase I observations is very large. The proble is that accurate estiation of an extree quantile in a nonparaetric way needs a huge nuber of observations. Replacing the nonparaetric individual chart by a nonparaetric chart for grouped observations does the job. These iproved data driven control charts are presented here. Ready-ade forulas are given, which ake ipleentation of the charts quite straightforward. An application on real data clearly shows the iproveent: estiation of extree quantiles is replaced by estiation of ordinary quantiles, which can be done in an accurate way for coon saple sizes. AMS Subject Classification: 62P30, 62G32, 62G30 Key Words: statistical process control, Phase II control liits, order statistics, unbiasedness, exceedance probability, nonparaetric, odel selection, in- Received: April 4, 2007 c 2007, Acadeic Publications Ltd. Correspondence author

2 424 W. Albers, W.C.M. Kallenberg iu control chart 1. Introduction As long as norality holds, the classical control chart for onitoring the ean of a production process can be applied very well. When norality fails as it unfortunately often does in the tails of the underlying distribution quantities like the false alar rate (FAR or the average run length (ARL ay be far away fro their prescribed values. Model errors of ore than 400% are not uncoon, see e.g. Chan et al [13], Pappanastos and Adas [15], Albers et al [5, 6]. To avoid these errors as a first step the faily of noral distributions can be ebedded in a larger paraetric odel. The so called noral power faily is a very useful and natural faily for this purpose. It has an additional shape paraeter, which is intiately related to the construction of control charts. This can be seen as follows. Control liits are deterined by quantiles of the underlying distribution. An upper liit (UL and a lower liit (LL are selected and an out-of-control (OoC signal is produced as soon as a newly arriving easureent falls outside these liits. In order to get a prescribed FAR like 0.002, UL and LL are chosen as the upper and lower quantiles of the underlying distribution. Essentially the quantiles of the standardized noral power faily are powers of the quantiles of the standard noral distribution. By taking a power larger than one heavier tails are odelled, while powers saller than one result in lighter tails than the noral distribution. By taking different shape paraeters at the lower and upper control liit, skewness of the distribution can be dealt with as well. Another paraetric faily with a siilar flavor is the Burr distribution, see e.g. Yourstone and Zier [20]. Typically, the paraeters of the proposed distribution are not known. Therefore, Phase I observations X 1,...,X n need to be used to estiate the paraeters. The price we have to pay for the extension to a larger paraetric faily is a larger stochastic error due to estiation of two extra paraeters, one for the lower and one for the upper control liit. When the paraetric faily still is too restricted to odel the underlying distribution accurately enough, a nonparaetric control chart coes into the picture. This topic by now has also received considerable attention in the literature, see e.g. Bakir and Reynolds [9], Bakir [7, 8], Chakraborti et al [11, 12], Chakraborti and Eryilaz [10], Qiu and Hawkins [17, 18] and Albers and Kallenberg [1]. However, applying a nonparaetric control chart ay

3 IMPROVED DATA DRIVEN CONTROL CHARTS 425 result in an unacceptably large stochastic error. For instance, to construct the upper liit for a one-sided control chart (with group size = 1 such that FAR = 0.001, we need to estiate the quantile. Obviously, with e.g. 100 observations this cannot be done in a truly nonparaetric way. In Albers et al [5] the three control charts (noral, paraetric, nonparaetric were cobined by a data driven choice between the. This cobined control chart works very well in any situations. The just entioned proble what to do when the data tell us to apply the nonparaetric chart and the nuber of observations is not very large, was partly solved there by introducing a odified nonparaetric control chart. The nonparaetric chart itself ay lead to a randoized upper control liit, which equals with substantial probability. Although being theoretically correct, such a chart which ay result in never producing an out-of-control signal is obviously totally unacceptable fro a practical point of view. Therefore, in the odified version, the control liit was substituted by a very large quantity, in fact X (n + S, where X (n denotes the largest observation fro the Phase I observations X 1,...,X n and S its saple standard deviation. First of all, this odified version is not really a nonparaetric control chart. Moreover, for saple sizes n 250, the expected F AR occasionally turned out to be twice as large as it should be. Therefore, iproveent of this part of the procedure should be very welcoe. Such an iproveent can be obtained by considering at this stage a soewhat larger group size. It has been shown in Albers and Kallenberg [4] that, surprisingly, rather than using the group averages, it is definitely preferable to copare the iniu for each group to a suitably chosen upper control liit and, siilarly, the axiu to an analogous lower control liit. We call such a chart a iniu control chart or shortly a M IN chart. The reason for the preference of a MIN chart is that in the nonparaetric case the saple ean or average turns out to be neither optial nor easy to work with. Outside the noral odel exaples are easily found where M IN outperfors the average chart, while under norality the M IN chart behaves rather well and is substantially better than the individual chart for interesting shifts under outof-control. More iportantly for the present purpose, the estiation step for the average chart still requires uncofortably large values of n. This sharply contrasts the situation with the MIN chart, since for that chart a uch less extree quantile has to be estiated. For exaple, for = 3 and p = the (p 1/ = quantile coes in. For ore details we refer to Albers and Kallenberg [4].

4 426 W. Albers, W.C.M. Kallenberg This new nonparaetric control chart opens the possibility to fill the gap in the data driven control chart when n is not huge and the selection rule points towards the nonparaetric chart. In this paper the data driven control chart is iproved by adding at this stage the MIN chart. The paper is organized as follows. When estiation coes in, quantities like FAR and ARL becoe rando variables, as they are functions of the Phase I observations X 1,...,X n. In Section 2 criteria are given to express how far away the stochastic F AR and ARL are fro their prescribed values. In particular, the bias and the so called exceedance probability are considered. Moreover, the concept of average tie to signal (ATS is introduced, which relates the individual chart with a chart based on groups. The iproved data driven control chart is presented in Section 3. The three building blocks, the noral, the paraetric and the MIN chart, are described, as well as the selection rule to choose between the. The ready-ade forulas ake ipleentation, for instance in Excel, quite straightforward. Therefore, the proposed ethods are very accessible for practitioners. In an application to real data in Section 4 it is shown how the iproved data driven control chart works in practice. It is seen that for the upper liit the classical noral chart can be used, while the lower liit requires a nonparaetric chart, due to the heavy lower tail. The individual nonparaetric chart akes no sense here, while the MIN chart perfors very well. A brief suary including theoretical results is given in Section Criteria A one-sided control chart with an upper control liit such that FAR = p is easily obtained when the continuous distribution F of the observations under incontrol (IC is known. One siply takes UL = F 1 (p, where F(x = 1 F(x and hence F 1 (p is the upper p-quantile of X, having distribution function F. For instance, when X is norally distributed with known ean µ and variance σ 2 we get UL = µ+u p σ with u p = Φ 1 (p, the upper p-quantile of the standard noral distribution. Often one chooses u p = 3 corresponding to FAR = p = and ARL = 1/p = 741. (Note that we assue independence of the observations and hence do not include autocorrelated tie series. When the paraeters of F (in the noral or paraetric control chart are unknown or F itself (in the nonparaetric control chart is unknown, we have to estiate the upper p-quantile using Phase I observations X 1,...,X n, which are

5 IMPROVED DATA DRIVEN CONTROL CHARTS 427 assued to be IC. As a consequence, FAR and ARL are no longer deterinistic quantities, but rando variables, because they depend on X 1,...,X n. The idea is that the stochastic FAR, written as P n or the stochastic ARL = 1/P n should be close to the prescribed p or 1/p, respectively. Several criteria can be used for it. We ay focus on the average behavior of the chart during a long series of separate applications by concentrating on the bias EP n p (we do not consider the bias of the stochastic ARL, since E (1/P n does not adequately suarize the run length properties of the chart, cf. e.g. Roes [19, p. 34], Quesenberry [16, p. 242] and Jones et al [14, p. 100]. On the other hand we want to see how bad things can get in one given application of the chart. Then we look at the so called exceedance probabilities like P(P n > p(1 + ε with ε sall. They should be saller than α, say, a second sall quantity. The idea is that the probability that P n is substantially larger than p (that is larger than p(1+ε is sall: less than α. One ay think of values for α and ε like 0.1 or 0.2. To get a sall bias or sall exceedance probabilities, huge saples are needed when estiators of the paraeters are siply plugged in into the control liits. Fortunately, corrections of the control liits are available reducing the bias and exceedance probabilities sufficiently well for coon saple sizes, except for the nonparaetric control charts. Indeed, as explained in the introduction, in the nonparaetric case the stochastic error is so large that a truly nonparaetric chart would iply that with substantial probability no OoCsignal is given at all. The reedy for it is the application of the MIN chart based on groups of observations with group size. Since we are dealing with groups of observations we consider in that case the so called average tie to signal (ATS which equals /FAR and copare this for the IC-situation to the prescribed 1/p. Here FAR refers to the failure rate for the group of observations. So, when we have Phase II observations X n+1,...,x n+ and UL denotes the upper control liit of the MIN chart, we get (in the one sided case FAR = P (in (X n+1,...,x n+ > UL. In practice UL should be estiated by ÛL, say, and (the corrected ÛL is chosen in such a way that (1/ ties the observed F AR has a sall bias w.r.t. p or that its exceedance probability is sall. For the two-sided control charts we siply cobine the upper and lower control charts, replacing p by p/2. The observed F AR s for the two-sided control charts are defined as follows. For the noral and paraetric chart we have P nl = P (X n+1 < LL X 1,...,X n

6 428 W. Albers, W.C.M. Kallenberg with LL = LL N, say, for the noral chart and LL = LL P for the paraetric chart and P nu = P (X n+1 > ÛL X 1,...,X n with ÛL = ÛL N for the noral chart and ÛL = ÛL P for the paraetric chart. For the nonparaetric MIN chart we define P nl = P (ax(x n+1,...,x n+ < LL MIN X 1,...,X n and P nu = P In all cases we have (in (X n+1,...,x n+ > ÛL MIN X 1,...,X n. P n = P nl + P nu. 3. Iproved Data Driven Control Chart To present the iproved data driven control chart we start with describing its building blocks: the noral chart, the paraetric chart and the iniu chart, the latter being a nonparaetric chart. They are obtained by starting with UL = F 1 (p and LL = F 1 (p for the noral and paraetric chart, and UL = F 1 (p 1/ and LL = F 1 (p 1/ for the iniu chart, thus giving F AR = p for all three charts. Next appropriate estiators are plugged in, based on Phase I observations X 1,...,X n. We write X = n 1 X i and S = S 2 with S 2 = (n 1 1 ( X i X 2. This gives the uncorrected control liits X ± u p/2 S in the noral case, X ± c( γu 1+bγ p/2 S in the paraetric case (see (3.1 and (3.2 for definition of c(γ and F 1 n (p 1/,Fn 1 (p 1/ for the iniu chart, where F n denotes the epirical distribution function based on X 1,...,X n. Finally correction ters are added according to the prescribed ai, to reduce the bias or the exceedance probability. Here we siply present the forulas. For ore inforation and derivation of these charts we refer to Albers and Kallenberg [2, 3, 4] and references therein. Although the derivations of the charts are rather coplicated, as to the coputational aspects of the control liits it should be rearked that the actual control liits are explicitly given here and they can be obtained quite straightforwardly, for instance using Excel, because all required quantities (like the inverse of the standard noral distribution function and the Gaa function are available in Excel.

7 IMPROVED DATA DRIVEN CONTROL CHARTS 429 Ai EP n = p Pr `P nl > p 2 (1 + ε α, Pr `P nu > p 2 (1 + ε α Pr Pr 1 P nl < 1 1 P nu < 1 p/2 (1 ε α, p/2 (1 ε α LLN, c ( UL d N X ± u p/2 S 1 + u2 p/ n 8 1/2 >< 1 u α + X ± u p/2 S >: u 2 p/2 ε n 8 >< u α X ± u p/2 S >: u 2 p/2 n 1/2 u 2 p/2 u 2 p/2 9 >= >; 9 >= ε (1 ε >; Table 1: Two-sided (corrected noral control liits 3.1. Noral Control Charts The two-sided control liits for the (corrected noral control chart are given in Table 1. The control liits are denoted by LL N and ÛL N, where the subscript N refers to norality. In case of the lower liit LL N the sign should be read and for ÛL N the + sign is used. An OoC-signal is given( when a new incoing Phase II observation X n+1, say, is outside the interval LLN N,ÛL. The one-sided versions are siply obtained by replacing p/2 in the control liits by p and taking the corresponding one-sided intervals Paraetric Control Charts To ebed the noral distributions in a larger faily with heavier or lighter tails we consider essentially powers of the standard noral quantiles as the new quantiles. The faily of distributions obtained in this way is called the noral power faily. More precisely, replace u p (for 0 < p < 1/2 by c(γu 1+γ p, (3.1 where γ > 1 and where c(γ is a noralizing constant (to ake the variance equal to one given by c(γ = π 1/4 2 (1+γ/2 Γ(γ /2 (3.2 with Γ the Gaa-function. It is iediately seen that γ = 0 leads to the standard noral quantile u p. Let X (1... X (n be the order statistics of X 1,...,X n. The estiator

8 430 W. Albers, W.C.M. Kallenberg of γ at the upper tail, γ U, is given by (ent(x is the integer part of x ( X(ent(0.95n+1 X γ U = log 1. X (ent(0.75n+1 X Note that ( X (ent(0.95n+1 X / ( X (ent(0.75n+1 X estiates ( ( / = (u 0.05 /u γ c(γu 1+γ 0.05 c(γu 1+γ 0.25 and that 1/log (u 0.05 /u 0.25 = Siilarly, the estiator of γ at the lower tail, γ L, is given by ( X X(n ent(0.95n γ L = log 1. X X (n ent(0.75n By taking estiators of γ according to the tail under consideration allowance is ade for possible skewness of the distribution. The uncorrected control liits are given by X ± c( γu 1+bγ S. Before presenting the corrected control liits we p/2 introduce soe notation, which is needed for the correction ters (see Table 2 : C 1 (γ,u p = γ γ u p 0.08γu p 0.14γ 2 u p, ( 1+γ uan C 2 (γ = γ u bn ent (0.95n + 1 ent (0.75n + 1 with a n = 1,b n = 1, n + 1 n + 1 C 3 (γ,u p = γ 81.93γ u p γu p γ 2 u p, A(γ,u p = γ 10.02γ u p γu p γ 2 u p. For a detailed explanation of these forulas we refer to Albers et al [5, 6]. The two-sided control liits for the (corrected paraetric control chart are given in Table 2. They are denoted by LL P and ÛL P, where the subscript P refers to paraetric. In case of the lower liit LL P the sign should be read and γ = γ L should be inserted, while for ÛL P the + sign is used and γ = γ U. An OoC-signal is given ( when a new incoing Phase II observation X n+1 is outside the interval LLP P,ÛL. The one-sided versions are siply obtained by replacing p/2 in the control liits by p and taking the corresponding onesided intervals.

9 IMPROVED DATA DRIVEN CONTROL CHARTS 431 Ai EP n = p Pr `P nl > p (1 + ε α, 2 Pr `P nu > p (1 + ε α 2 1 Pr P nl < 1 (1 ε α, p/2 1 Pr P nu < 1 (1 ε α p/2 j X ± S X ± S LLP, c UL d P c(bγu 1+bγ p/2 C1(bγ, u p/2c 2(bγ + C3(bγ, u ff p/2 j n c(bγu 1+bγ p(1+ε/2 + A(bγ, u ff p/2u α n j X ± S c(bγu 1+bγ p/{2(1 ε} + A(bγ, u ff p/2u α n Table 2: Two-sided (corrected paraetric control liits based on the noral power faily 3.3. Miniu Control Charts The two-sided control liits for the (corrected iniu control chart are given in Table 3. They are denoted by LL MIN and ÛL MIN, where the subscript MIN refers to iniu. An OoC-signal is given when for a new incoing group of Phase II observation X n+1,...,x n+ either in (X n+1,...,x n+ > ÛL MIN or ax (X n+1,...,x n+ < LL MIN. Before presenting the iniu control liits we introduce the following notation: ( r = ent n { (p/2} 1/, ( r k 1 + k is deterined by λ = ( n + (p/2 ( r k + ( r k 1 + ( r k 1 + ( n + (p/2, ( r k + < B(n,p,j = P(Z j,b(n,p,j = P(Z = j with Z binoial(n,p, q(p = (p 1/, k(p,α is deterined by B(n,q(p,r k(p,α 1 α λ(p,α = α B(n,q(p,r k(p,α 1 b(n,q(p,r k(p,,α k = k((1 + εp/2,α, λ = λ((1 + εp/2,α, < B(n,q(p,r k(p,α,,

10 432 W. Albers, W.C.M. Kallenberg Ai E Pn = p Pr PnL > p 2 (1 + ε α, Pr PnU > p (1 + ε α 2 Pr P nl < 1 (1 ε α, p/2 Pr P nu < 1 (1 ε α p/2 c LLMIN, d UL MIN c LLMIN : (1 λx (r k + λx (r+1 k dul MIN : (1 λx (n+k+1 r + λx (n+k r cll MIN : (1 e λx (r e k + e λx (r+1 e k dul MIN : (1 e λx (n+ e k+1 r + e λx (n+ e k r cll MIN : (1 λx (r k + λx (r+1 k dul MIN : (1 λx (n+k+1 r + λx (n+k r Table 3: Two-sided (corrected iniu control liits ( p k = k 2(1 ε,α ( p,λ = λ 2(1 ε,α 3.4. Selection Rule The idea behind the selection rule is to stay as long as possible in the classical noral chart, to ove to the paraetric chart if the tails are too heavy or too light and to take the nonparaetric MIN chart when the paraetric faily presuably fails too. The data are telling us which chart to use. Since control liits are deterined by the behavior in the far tail (p is sall, the natural yardstick to decide between the charts is the standardized axiu of the Phase I observations (X (n X/S for the upper liit and siilarly the standardized iniu ( X X (1 /S for the lower liit. In view of a possible odest outlier it ay see risky to use the (standardized largest and/or sallest observation in the selection rule. Quite the contrary is the case. When an outlier is present, we will take the ore safe nonparaetric iniu control chart, where the ost extree observations are not involved in deterining the control liits. Note that extree outliers ay have a asking effect, but such kind of outliers are not expected to be present, because the observations X 1,...,X n are authorized as being IC and hence nonconforing extree outliers should have been detected before. The next point is the deterination of the cut-off points. Distributions with heavier tails ay give serious probles with the IC behavior leading to invalid control charts. Distributions with thinner tails are conservative in the IC case with as consequence a loss in the OoC. Because IC-errors are ore serious than those in the OoC case and oreover, since a positive odel error as large as p or larger can easily occur, whereas the negative odel error is at ost p, we take the selection rules unbalanced. In particular, for the upper control liit

11 IMPROVED DATA DRIVEN CONTROL CHARTS 433 we will prefer the noral control chart when u ( log n/n X (n X u S 5/(n n. To see what this iplies, let X 1,...,X n be i.i.d. rando variables with a standard noral distribution. Then P ( ( log(n n X (n < u ( log n/n = 1 n exp ( log n 2, n and P (X (n > u 5/(n n ( = n ( 1 exp 5 5. n n n n Hence, when norality holds we stay with high probability at the noral chart. When the standardized axiu (X (n X/S is very large, this indicates that the tail ay be heavier than the noral one and we change to the paraetric or nonparaetric chart. Siilarly we change for relatively sall values of (X (n X/S, indicating a thinner tail behavior. That the latter is less severe, is expressed in the probability 2/ n, copared to 5/ n. The total two-sided iproved data driven control chart is given by the following schee. Let d 1N = u ( log n/n,d 2N = u 5/(n n, d 1P ( γ = c( γu 1+bγ ( log n/n,d 2P ( γ = c( γu 1+bγ 3/(n n and denote the lower and upper liit by LL C and ÛL C, respectively. The upper liit is obtained as follows d 1N X (n X yes S d 2N no d 1P ( γ U X (n X S d 2P ( γ U no yes ÛL C = ÛL N ÛL C = ÛL P ÛL C = ÛL MIN Here ÛL N,ÛL P and ÛL MIN are taken fro Tables 1 3 according to the required ai: bias, exceedance with FAR or exceedance with ARL or ATS. For the lower liit LL C we get

12 434 W. Albers, W.C.M. Kallenberg d 1N X X (1 yes S d 2N no d 1P ( γ L X X (1 S d 2P ( γ L no yes LL C = LL N LL C = LL P LL C = LL MIN Here LL N, LL P and LL MIN are taken fro Tables 1 3 according to the required ai: bias, exceedance with FAR or exceedance with ARL or ATS. 4. Application We apply the iproved data driven control chart on a real life exaple concerning the production of electric shavers by Philips. In an electrocheical process razor heads are fored. The easureents concern the thickness of these razor heads on a particular spot on the head. Available are two saples of 835 easureents each. In Albers et al [5] this data set has been utilized to illustrate the data driven approach of that paper with the whole first saple of 835 observations as Phase I observations. Instead of using the whole first saple we now take only 100 observations of the first saple, that is we have n = 100 here. The saple ean of this Phase I saple equals and the saple standard deviation is 3.07, while the sallest and largest observations are and 49.95, respectively. Hence, we obtain (x (n x/s = 2.33,(x x (1 /s = Direct calculation gives d 1N = 2.14 and d 2N = 2.58, iplying that the upper liit is given by the noral chart. Application of Table 1 with p/2 = 0.001,α = 0.2 and ε = 0.2 gives as control liits bias ÛL C = , exceedance FAR ÛL C = , exceedance ATS ÛL C = We apply the control charts on the reaining 1570 observations. Figure 1 shows the result, where the horizontal lines give the various upper control liits. Next we consider the lower part of the iproved data driven control chart. Since ( x x (1 /s = 3.83 > d2n = 2.58, the data tell us that the lower tail is too heavy to use the noral chart. Direct calculation gives γ L = and d 2P ( γ L = Because ( x x (1 /s = 3.83 > d2p ( γ L = 3.699, the

13 IMPROVED DATA DRIVEN CONTROL CHARTS 435 Figure 1: Iproved data driven control chart (upper part of the reaining 1570 easureents of the thickness of razor heads with ÛL C = (bias corrected, ÛL C = (exceedance FAR corrected and ÛL C = (exceedance ATS corrected paraetric chart cannot be used either and hence the nonparaetric M IN chart should be used at the lower part. We take = 3. As we have n = 100 and p/2 = 0.001, we therefore arrive at r = 14. This already shows that even for a value like = 3 no longer very extree order statistics coe in as with the individual chart, but ordinary ones. Indeed, the p/2 = upper quantile is replaced by the q = (p/2 1/ = upper quantile. While the upper quantile cannot be estiated nonparaetricly in a sensible way with 100 observations, the upper quantile can be estiated very well. This is the iproveent we were looking for. Since α = 0.2 and ε = 0.2, we get k = 1,λ = 0.72 for the bias correction, k = 2, λ = 0.74 for the exceedance case when correcting FAR and k = 2,λ = 0.95 for the exceedance correction w.r.t. ATS. In passing it is seen that indeed protection against exceedance effects requires a larger correction than against bias. Noting that x (12 = 39.09,x (13 = and x (14 = 39.82, it easily follows that the lower liits are given by bias LLC = , exceedance FAR LLC = , exceedance ATS LLC =

14 436 W. Albers, W.C.M. Kallenberg Figure 2: Iproved data driven control chart (lower part of the 523 triples contained in the reaining 1570 easureents of the thickness of razor heads with LL C = (bias corrected, LLC = (exceedance FAR corrected and LL C = (exceedance ATS corrected We apply the control charts on the 523 triples contained in the reaining 1570 observations. Figure 2 shows the result, where the horizontal lines give the several upper control liits. 5. Conclusion The open end in the data driven control chart, when the nuber of Phase I observations is not huge and a nonparaetric approach is chosen by the data, is filled in with the iproved data driven control chart. The application in Section 4 clearly shows the iproveent. When we would have used the truly nonparaetric control chart in the lower liit, this liit would be with probability 0.9 and x (1 = with probability 0.1 (in the bias case. Obviously, such a rule can never be recoended in practice: with very high probability there will be never a signal at all! The odified nonparaetric control chart gives soe kind of reedy, but is not a truly nonparaetric chart. In the application of Section 4 it gives as lower liit: x (1 s = with probability 0.9 and x (1 = with probability 0.1. The liit turns out to cause 3 ties a OoC-signal, while results in 9 signals.

15 IMPROVED DATA DRIVEN CONTROL CHARTS 437 In Albers et al [5] the theoretical behavior of the data driven control chart is studied in detail. It has been shown there that the behavior of the data driven control chart is asyptotically equivalent to the behavior of each specific control chart in its own doain. Since these results are a consequence of the selection rule and the selection rule in the iproved data driven control chart is the sae, these theoretical results apply for the iproved data driven control chart as well. Hence, in contrast to for instance the classical noral chart, the iproved data driven control chart is valid for all distributions. In each of the three situations (norality, noral power faily, outside the noral power faily its out-ofcontrol behavior is asyptotically as good as that of the specific corresponding control chart. For ore details of the proof and precise theoretical stateents we refer to Albers et al [5]. References [1] W. Albers, W.C.M. Kallenberg, Epirical nonparaetric control charts: estiation effects and corrections, Journal of Applied Statistics, 31 (2004, [2] W. Albers, W.C.M. Kallenberg, Self adapting control charts, Statistica Neerlandica, 60 (2006, [3] W. Albers, W.C.M. Kallenberg, Control charts in new perspective, Sequential Analysis, (2007, To Appear. [4] W. Albers, W.C.M. Kallenberg, Miniu control charts, Journal of Statistical Planning and Inference, (2007, To Appear. [5] W. Albers, W.C.M. Kallenberg, S. Nurdiati, Paraetric control charts, Journal of Statistical Planning and Inference, 124 (2004, [6] W. Albers, W.C.M. Kallenberg, S. Nurdiati, Exceedance probabilities for paraetric control charts, Statistics, 39 (2005, [7] S.T. Bakir, A distribution-free Shewhart quality control chart based on signed-ranks, Quality Engineering, 16 (2004, [8] S.T. Bakir, Distribution-free quality control charts based on signed-ranklike statistics, Counications in Statistics: Theory and Methods, 35 (2006,

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