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1 Ths artcle was downloaded by: [UVA Unverstetsbblotheek SZ] On: 16 May 212, At: 6:32 Publsher: Taylor & Francs Informa Ltd Regstered n England and Wales Regstered Number: Regstered offce: Mortmer House, Mortmer Street, London W1T 3JH, UK Technometrcs Publcaton detals, ncludng nstructons for authors and subscrpton nformaton: A Robust Standard Devaton Control Chart Mart Schoonhoven a & Ronald J.M.M. Does a a Insttute for Busness and Industral Statstcs, Unversty of Amsterdam, 118 TV, Amsterdam, The Netherlands Avalable onlne: 2 Dec 211 To cte ths artcle: Mart Schoonhoven & Ronald J.M.M. Does (212): A Robust Standard Devaton Control Chart, Technometrcs, 54:1, To lnk to ths artcle: PLEASE SCROLL DOWN FOR ARTICLE Full terms and condtons of use: Ths artcle may be used for research, teachng, and prvate study purposes. Any substantal or systematc reproducton, redstrbuton, resellng, loan, sub-lcensng, systematc supply, or dstrbuton n any form to anyone s expressly forbdden. The publsher does not gve any warranty express or mpled or make any representaton that the contents wll be complete or accurate or up to date. The accuracy of any nstructons, formulae, and drug doses should be ndependently verfed wth prmary sources. The publsher shall not be lable for any loss, actons, clams, proceedngs, demand, or costs or damages whatsoever or howsoever caused arsng drectly or ndrectly n connecton wth or arsng out of the use of ths materal.

2 Supplementary materals for ths artcle are avalable onlne. Please go to A Robust Standard Devaton Control Chart Mart SCHOONHOVEN and Ronald J.M.M. DOES Insttute for Busness and Industral Statstcs, Unversty of Amsterdam 118 TV Amsterdam, The Netherlands (m.schoonhoven@uva.nl; r.j.m.m.does@uva.nl) Ths artcle studes the robustness of Phase I estmators for the standard devaton control chart. A Phase I estmator should be effcent n the absence of contamnatons and resstant to dsturbances. Most of the robust estmators proposed n the lterature are robust aganst ether dffuse dsturbances, that s, outlers spread over the subgroups, or localzed dsturbances, whch affect an entre subgroup. In ths artcle, we compare varous robust standard devaton estmators and propose an algorthm that s robust aganst both types of dsturbances. The algorthm s ntutve and s the best estmator n terms of overall performance. We also study the effect of usng robust estmators from Phase I on Phase II control chart performance. Addtonal results for ths artcle are avalable onlne as Supplementary Materal. KEY WORDS: 1. INTRODUCTION The performance of a process depends on the stablty of ts locaton and dsperson parameters and any change n these parameters should be detected as soon as possble. To montor these parameters, Shewhart (1931) ntroduced the dea of control charts n the 192s. The dsperson parameter s controlled frst, followed by the locaton parameter. The present artcle focuses on control charts for montorng the standard devaton. We assume that n the desgn of such charts, the n-control standard devaton (σ ) s unknown. Therefore, σ must be estmated from subgroups taken when the process s assumed to be n control. Ths stage n the control chartng process s denoted as Phase I (cf. Woodall and Montgomery 1999). Control lmts are calculated from the estmated σ to montor the process standard devaton n Phase II. The Phase I and Phase II data are arranged n subgroups ndexed by. We denote by X j, = 1, 2,...,kand j = 1, 2,...,n, the Phase I data and by Y j, = 1, 2,... and j = 1, 2,...,n, the Phase II data. The X j s are assumed to be ndependent and N(µ, σ 2 ) dstrbuted and the Y j s are assumed to be ndependent and N(µ, (λσ ) 2 )dstrbuted, where λ s a constant. When λ = 1, the standard devaton s n control; otherwse t has changed. Let ˆσ be an unbased estmate of σ based on the X j s, and let ˆσ be an unbased estmate of λσ based on the th subgroup Y j, j = 1, 2,...,n. The process standard devaton can be montored n Phase II by plottng ˆσ on a Shewhart-type control chart wth lmts Average run length; Mean squared error; Phase I; Phase II; Statstcal process control. ÛCL = U n ˆσ, LCL = L n ˆσ, (1) where U n and L n are chosen so that the desred control chart behavor s acheved when the process s n control. When ˆσ falls wthn the control lmts, the process s deemed to be n control. We defne F as the event that ˆσ falls beyond the lmts, P (F ) = p as the probablty of that event and RL as the run length, that s, the number of subgroups untl the frst ˆσ falls beyond the lmts. When the lmts are known, F s and F t (s t) are ndependent and therefore RL s geometrcally dstrbuted wth parameter p. Hence, the average run length (ARL) s gven by 1/p and the standard devaton of the run length (SDRL) by 1 p/p. It s common practce to use p =.27 and so ARL = 37.4 and SDRL = when σ s known. When the standard devaton s estmated, the condtonal run length the run length gven an estmate of σ has a geometrc dstrbuton. However, the uncondtonal RL dstrbuton the run length dstrbuton averaged over all possble values of the estmated σ s not geometrc. Quesenberry (1993) showed that for the X and X control charts, the uncondtonal ARL as well as the uncondtonal p are hgher than n the (µ,σ )-known case. Chen (1998) studed the uncondtonal run length dstrbuton of the standard devaton control chart and showed that the stuaton s somewhat better than for the X control chart. To acheve the ntended uncondtonal n-control performance when the lmts are estmated, one could derve U n and L n by controllng ether the n-control p or ARL, or a percentle pont of the n-control RL dstrbuton. An advantage of usng the ARL s ts ntutve nterpretaton. A drawback, however, s that the ARL s strongly determned by the occurrence of extremely long runs. Hller (1969) and Yang and Hller (197) derved correcton factors for the range (R) and standard devaton (S) control charts by controllng p. Jensen et al. (26) conducted a lterature survey of the effects of parameter estmaton on control chart propertes and dentfed ssues for future research. Ther suggeston on page 36 s the subject of the present artcle. More specfcally, we wll fnd robust estmators for Phase I data and we wll study the performance of these robust estmators durng Phase II montorng. Rocke (1989) proposed standard devaton control charts based on the mean or the trmmed mean of the subgroup ranges or subgroup nterquartle ranges. Moreover, he studed a twostage procedure whereby the ntal chart s constructed frst and then groups that seem to be out of control are excluded. The control lmts are recomputed from the remanng subgroups. Rocke (1992) provded the practcal detals for the 212 Amercan Statstcal Assocaton and the Amercan Socety for Qualty DOI: 1.18/

3 74 MARIT SCHOONHOVEN AND RONALD J.M.M. DOES constructon of these charts. Tatum (1997) explaned the dfference between dffuse and localzed dsturbances: dffuse dsturbances are equally lkely to perturb any observaton, whereas localzed dsturbances affect all observatons n a subgroup. He proposed a method, constructed around a varant of the bweght A estmator, that s resstant to both dffuse and localzed dsturbances. Fnally, Davs and Adams (25) proposed a dagnostc technque for montorng data that mght be contamnated wth outlers to react to sgnals that ndcate a true process shft only. In ths artcle, we nvestgate robust Phase I estmators for the subgroup standard devaton control chart. The estmators consdered are the pooled standard devaton, the robust bweght A estmator of Tatum (1997), and several adaptve trmmers. Addtonally, we look at an adaptve trmmer based on the mean devaton from the medan, a statstc more resstant to dffuse outlers (cf. Schoonhoven, Raz, and Does 211). For dffuse outlers, we thnk that a control chart for ndvdual observatons would detect outlers more quckly. We therefore nclude an estmator based on the ndvduals chart. To measure the varablty wthn and not between subgroups, we correct for dfferences n the locaton between the subgroups. Fnally, we present an algorthm that combnes the last two approaches. The performance of the estmators s evaluated by assessng ther mean squared error () under normalty and n the presence of several types of contamnatons. Moreover, we derve factors for the Phase II lmts of the standard devaton control chart and assess the performance of the control charts by means of a smulaton study. The artcle s structured as follows. The next secton ntroduces the standard devaton estmators, demonstrates the mplementaton of the estmators by means of a real-world example, and assesses ther. Next, we present the desgn schemes for the standard devaton control chart and derve the Phase II control lmts. We then descrbe the smulaton procedure and smulaton results. The artcle ends wth some concludng remarks. 2. PROPOSED PHASE I ESTIMATORS In practce, the same statstc s generally used to estmate both the n-control standard devaton σ n Phase I and the standard devaton λσ n Phase II. Snce the requrements for the estmators dffer between the two phases, ths s not always the best choce. In Phase I, an estmator should be effcent n uncontamnated stuatons and robust aganst dsturbances, whereas n Phase II, the estmator should be senstve to dsturbances (cf. Jensen et al. 26). In ths secton, we present sx Phase I estmators, demonstrate the mplementaton of the estmators by means of a real data example, and assess the effcency of the estmators n terms of ther. 2.1 Estmators of the Standard Devaton Recall that X j, = 1, 2,...,k and j = 1, 2,...,n, denotes the Phase I data wth n the subgroup sze and k the number of subgroups. The frst estmator of σ s based on the pooled subgroup standard devaton ( 1 k ) 1/2 = S 2, (2) k where S s the th subgroup standard devaton defned by ( 1 n ) 1/2 S = (X j X ) 2. n 1 =1 j=1 An unbased estmator s gven by /c 4 (k(n 1) + 1), where c 4 (m) s defned by ( ) 2 1/2 Ɣ(m/2) c 4 (m) = m 1 Ɣ((m 1)/2). Ths estmator provdes a bass for comparson under normalty when no contamnatons are present. Mahmoud et al. (21) showed that ths estmator s more effcent than the mean of the subgroup standard devatons and the mean of the subgroup ranges when the data are normally dstrbuted. We also evaluate a robust estmator proposed by Tatum (1997). Ths approach s applcable for n 4. The method begns by calculatng the resduals n each subgroup, whch nvolves subtractng the subgroup medan from each value: res j = X j M.Ifn s odd, then n each subgroup, one of the resduals wll be zero and s dropped. As a result, the total number of resduals s m = nk when n s even and m = (n 1)k when n s odd. Tatum s estmator s gven by ( Sc = m k ( ) =1 j: u j <1 res2 j 1 u 2 4 ) 1/2 j (m 1) 1/2 k ( )( ), =1 j: u j <1 1 u 2 j 1 5u 2 j (3) where u j = h res j /(cm ), M s the medan of the absolute values all resduals, 1 E 4.5, h = E <E 7.5, c E > 7.5, and E = IQR /M.IQR s the nterquartle range of subgroup and s defned as the dfference between the second-smallest and the second-largest observaton for 4 n 7, and as the dfference between the thrd-smallest and the thrd-largest observaton for 8 n 11. The constant c s a tunng constant. Each value of c leads to a dfferent estmator. Tatum studed the behavor of the estmator for c = 7 and c = 1 and showed that c = 7 gves an estmator that loses some effcency n the absence of dsturbances but gans effcency n the presence of dsturbances. We apply ths value of c n our smulaton study. Note that we have h() = E 3.5 for4.5 <E 7.5 nstead of h() = E 4.5, as presented by Tatum (1997, p. 129). Ths was a typographcal error, resultng n too much weght on localzed dsturbances and thus an overestmaton of σ. An unbased estmator of σ s gven by Sc /d (c, n, k), where d (c, n, k) s a normalzng constant. Durng the mplementaton of the estmator, we dscovered that for odd values of n, the values of d (c, n, k) gven by Table 1 n Tatum (1997) are ncorrect. We use the corrected values, whch are presented n Table 1. The resultng estmator s denoted by, as n Tatum (1997).

4 A ROBUST STANDARD DEVIATION CONTROL CHART 75 Table 1. Normalzng constants d (c, n, k) for Tatum s estmator (S c ) c = 7 c = 1 n k = 2 k = 3 k = 4 k = 2 k = 3 k = We also nclude other procedures to obtan ˆσ. The frst s a varant of Rocke (1989). Rocke s procedure frst estmates σ by the mean subgroup range R = 1 k k R, (4) =1 where R s the range of the th subgroup. An unbased estmator of σ under normalty s R/d 2 (n), where d 2 (n) sthe expected range of a random N(, 1) subgroup of sze n. Values of d 2 (n) can be found n Duncan (1974, table M). Any subgroup that exceeds the Phase I control lmts s deleted and R s recomputed from the remanng subgroups. Our approach s smlar but contnues untl all subgroup ranges fall between the Phase I control lmts. These are set at ÛCL = U n R/d 2 (n) and LCL = L n R/d 2 (n). We derve the factors U n and L n from the and.135 quantles of the dstrbuton of R/d 2 (n). Table 2 shows the factors for n = 4, 5, 9 as well as the constants added to obtan unbased estmates from the screened data. The factors as well as the constants are obtaned by smulaton. Note that the factors and the constants are the same for k = 2, 5, 1. The resultng estmator s denoted by R s. In addton, we evaluate an adaptve trmmer where the estmate of σ s obtaned by the mean subgroup average devaton from the medan nstead of R. The mean subgroup average devaton from the medan s gven by MD = 1 k k MD, (5) where MD s the average absolute devaton from the medan M of subgroup defned by n MD = X j M /n. j=1 =1 An unbased estmator of σ s MD/t 2 (n), where t 2 (n) equals E(MD/σ ). Snce t s dffcult to obtan E(MD) analytcally, t s obtaned by smulaton. Extensve tables for t 2 (n) can be found n Raz and Saghr (29). The advantage of ths estmator s that t s less senstve to outlers than R (cf. Schoonhoven et al. 211). The resultng estmator s denoted by MD s. The values used for the Phase I control lmts and the constants necessary to obtan unbased estmates from the screened data are gven n Table 2. Both are obtaned by smulaton. For subgroup control charts, only adaptve trmmng methods based on the subgroup averages or subgroup standard devatons have been proposed n the lterature so far. For dffuse outlers, however, an ndvduals chart should detect outlers more quckly. We therefore propose a screenng method based on an ndvduals chart. The algorthm frst calculates the resduals by subtractng the subgroup medan from each observaton n the correspondng subgroup. Ths ensures that the varablty s measured wthn and not between subgroups. Next, an ndvduals chart of the resduals s constructed. The locaton of the chart (µ) s estmated by the mean of the subgroup medans, whch s zero because the subgroup medans have been subtracted from the observatons, and σ s estmated by MD. For smplcty, the factors for the ndvduals chart are 3 and 3 (see Table 2). The resduals that fall outsde the control lmts are excluded from the dataset. Then the procedure s repeated: the medan values of the adjusted subgroups are determned, the resduals are calculated, and the control lmts of the ndvduals chart are computed. The resduals that now exceed the lmts are removed. Ths contnues untl all resduals fall wthn the control lmts. Smulaton revealed that the resultng estmates of σ are slghtly based under normalty. The constants necessary to obtan an unbased estmate can be found n Table 2.The unbased estmator s denoted by MD. The above procedure does not use the spread of the subgroups. Therefore, we fnally propose an algorthm that combnes the use of an ndvduals chart wth subgroup screenng. Frst, an ntal estmate of σ s obtaned va MD (see (5)). Ths estmate s then used to construct a standard devaton control chart so that the subgroups can be screened. Adoptng R as a chartng statstc wll result n the excluson of many subgroups, ncludng many uncontamnated observatons, when dffuse dsturbances are present. For ths reason, we employ IQR for screenng purposes. The constants requred to obtan an unbased estmate of σ based on IQR are.594 for n = 4,.99 for n = 5, and for n = 9. The values chosen for the Phase I control lmts are presented n Table 2. The subgroup screenng s contnued untl all IQRs fall wthn the lmts. The resultng estmates of σ are Table 2. Factors for Phase I control lmts for k = 2, 5, 1 n = 4 n = 5 n = 9 Chart U n L n Constant U n L n Constant U n L n Constant R s MD s MD

5 76 MARIT SCHOONHOVEN AND RONALD J.M.M. DOES Table 3. Melt ndex measurements Sample Observatons R/d 2 (4) S/c 4 (4) MD/t 2 (4) IQR/d IQR (4) unbased and are used to screen observatons wth an ndvduals control chart (the procedure used to derve MD ). Smulaton revealed that the fnal estmates of σ are slghtly based. The constants necessary to obtan an unbased estmate can be found n Table 2. The unbased estmator s denoted by. 2.2 Real Data Example In ths secton, we demonstrate the estmaton of σ n Phase I. Our dataset was suppled by Wadsworth, Stephens, and Godfrey (21, pp ). The operaton concerns the melt ndex of a polyethylene compound. The data consst of 2 subgroups of sze4(table 3). The factors used for the n = 4,k = 2 case are presented n Table 2. Note that d 2 (4) = 2.6, c 4 (4) =.92, t 2 (4) =.66, and d IQR (4) =.59. The estmates of σ obtaned by and are determned n one teraton and are 1.14 and 6.59, respectvely. The values obtaned by R s and MD s ncorporate subgroup screenng. The ntal value of R s s 8.96 and the respectve upper and lower control lmts are 2.8 and The unbased estmate of the range (.e., R/d 2 (4)) of subgroup 3 falls above the control lmt and so ths subgroup s deleted. The second estmate of R s equals 7.92 and the correspondng Phase I upper and lower control lmts are and Now subgroup 4 does not meet the Phase I upper control lmt and s removed. The thrd estmate of R s s 7.31 and the control lmts are and There are no further subgroups whose R/d 2 (4) exceeds the upper control lmt. The resultng unbased estmate of σ s The MD s procedure works n a smlar way. In ths case, subgroups 3 and 4 are agan deleted. The fnal unbased estmate s 7.3. For the MD chart, we use a procedure based on the ndvduals control chart for the resduals. The resduals are calculated by subtractng the subgroup medan from each observaton n the correspondng subgroup (see Table 4). The ntal value of σ s 8.26 and the control lmts of the ndvduals chart are and One resdual n subgroup 3 and one resdual n subgroup 4 fall outsde the control lmts. The correspondng observatons are deleted from the dataset. The subgroup medans are determned from the remanng observatons and the resduals are recalculated. The second estmate of σ s 6.82 and the control lmts are now 2.47 and One resdual n subgroup 6 falls below the lower control lmt and so one observaton s removed. Agan, the medans are determned from the remanng observatons and the resduals are recomputed. The thrd estmate of σ s 6.49 and the control chart has lmts at and There are now no resduals that fall outsde the control lmts. The resultng unbased estmate s Sample Table 4. Resduals of melt ndex measurements Resduals

6 A ROBUST STANDARD DEVIATION CONTROL CHART 77 Table 5. Summary of estmates of σ and data deletons Chart ˆσ Deleted subgroup Deleted observaton R s ; 4 MD s 7.3 3; 4 MD :1; 4:1; 6: ; 7; 19 4:1; 6:1 For the chart, the frst part of the procedure screens the subgroup IQR. The respectve upper and lower control lmts of the IQR chart are and.15. The IQR of subgroups 3, 7, and 19 are and so these subgroups are deleted. It s not necessary to delete any further subgroups. Next, ndvdual observatons are screened. The estmate of σ s 7.81 and the upper and lower control lmts for the resduals are and An outler n subgroup 4 s deleted. The next estmate of σ s 7.18 wth correspondng control lmts and The outler n subgroup 6 s removed from the dataset. Now σ s set at 6.79 wth correspondng control lmts 2.37 and No further deletons are requred. The unbased estmate for the chart s The fnal estmates for σ as well as the data deletons are presented n Table 5. The estmate based on s hgher than the other estmates. Ths s because s more senstve to outlers than the other estmators. Note, however, that the queston of whch estmator has done the best job cannot be resolved from such a lmted dataset. 2.3 Effcency of the Proposed Estmators To evaluate Phase I performance, we now assess the of the proposed Phase I estmators. The s estmated as = 1 N N (ˆσ σ ) 2, (6) =1 where ˆσ s the value of the unbased estmate n the th smulaton run and N s the number of smulaton runs. Comparsons are made under normalty and four types of dsturbances (cf. Tatum 1997), but wth an error rate of 6% n each case. In general, we expect that a hgher error rate would result n more pronounced dfferences between the estmators. The four dsturbances are captured n: 1. A model for dffuse symmetrc dsturbances n whch each observaton has a 94% probablty of beng drawn from the N(, 1) dstrbuton and a 6% probablty of beng drawn from the N(,a) dstrbuton, wth a = 1.5, 2.,...,5.5, A model for dffuse asymmetrc varance dsturbances n whch each observaton s drawn from the N(, 1) dstrbuton and has a 6% probablty of havng a multple of a χ1 2 varable added to t, wth the multpler equal to.5, 1.,...,4.5, A model for localzed varance dsturbances n whch all observatons n three (when k = 5) or sx (when k = 1) subgroups are drawn from the N(,a) dstrbuton, wth a = 1.5, 2.,...,5.5, A model for dffuse mean dsturbances n whch each observaton has a 94% probablty of beng drawn from the N(, 1) dstrbuton and a 6% probablty of beng drawn from the N(b, 1) dstrbuton, wth b =.5, 1.,...,9., 9.5. The s obtaned for n = 5, 9 and k = 5, 1. The number of smulaton runs N s equal to 5,. Note that Tatum (1997) used 1, smulaton runs. Below we only present the results for k = 5 because the conclusons for k = 1 are very smlar. The fgures comparng k = 5 and k = 1 are avalable as Supplementary Materal. Fgure 1 shows the values when dffuse symmetrc varance dsturbances are present. The y-ntercepts show that the pooled standard devaton ( ) has the lowest when no dsturbances are present. However, when the sze of the dsturbance (a) ncreases, the ncreases quckly. The other estmators are more robust aganst outlers of ths type. Those that use an a (a) a (b) Fgure 1. of estmators when symmetrc dffuse varance dsturbances are present: (a) n = 5,k = 5 and (b) n = 9,k = 5.

7 78 MARIT SCHOONHOVEN AND RONALD J.M.M. DOES Multpler (a) Multpler (b) Fgure 2. of estmators when asymmetrc dffuse varance dsturbances are present: (a) n = 5,k = 5 and (b) n = 9,k = 5. ndvduals control chart to dentfy ndvdual outlers, that are, MD and, concde and perform best, followed by. The estmators based on only subgroup screenng, namely R s and MD s, turn out to perform less well n ths stuaton. The reason s that they screen subgroup dsperson and gnore ndvdual outlers. Note that R s falls far short of MD s, because R s uses R (rather than MD) to estmate σ.as R s more senstve to outlers, the Phase I lmts are broader, makng t more dffcult to detect outlers. Ths effect s partcularly sgnfcant for n = 9, because a larger subgroup s more lkely to be nfected wth an outler. When asymmetrc dffuse dsturbances are present (Fgure 2), the results are comparable to the stuaton wth dffuse symmetrc dsturbances: MD and concde and perform best, followed by and MD s. Note that n ths stuaton, and, for n = 9, R perform badly Fgure 3 shows the results n stuatons wth localzed dsturbances. The estmators ncorporatng subgroup screenng ( R s and MD s ) perform best. The estmator performs better than n ths stuaton. Fnally, MD does not perform as well n ths case because t does not take nto account nformaton on the subgroup spread. The results for the fourth type of dsturbance are shown n Fgure 4. We can conclude that and R s concde for n = 9 and perform far worse than the other estmators. MD s performs better but not as well as and not as well as the estmators usng an ndvduals chart to dentfy ndvdual outlers. The reason s that MD s s less capable of detectng such outlers. The estmators and MD concde and perform best n ths stuaton. Out man concluson from the above results s that the estmator performs best overall a (a) a (b) Fgure 3. of estmators when localzed varance dsturbances are present: (a) n = 5,k = 5 and (b) n = 9,k = 5.

8 A ROBUST STANDARD DEVIATION CONTROL CHART b (a) b Fgure 4. of estmators when dffuse mean dsturbances are present: (a) n = 5,k = 5 and (b) n = 9,k = DERIVATION OF THE PHASE II CONTROL LIMITS Equaton (1) gves control lmts for the standard devaton control chart wth σ estmated n Phase I. We estmate λσ n Phase II by S/c 4 (n) for all charts. One of the crtera used to assess Phase II performance s the ARL. To allow comparson, U n and L n are chosen such that the uncondtonal ARL equals 37 and, for each chart, the ARLs for the upper and lower control lmts are smlar. U n and L n cannot be obtaned easly n analytc form and are obtaned from 5, smulaton runs. Table 6 presents U n and L n for n = 5, 9 and k = 5, CONTROL CHART PERFORMANCE We now evaluate the effect of the proposed estmators on the Phase II performance of the standard devaton control chart. We consder the same Phase I estmators as those used to asses Table 6. Factors U n and L n to determne Phase II control lmts for n = 5andn = 9 k = 5 k = 1 n ˆσ U n L n U n L n R s MD s MD R s MD s MD the wth a,b and the multpler equal to 4 to smulate the contamnated cases (see Secton 2.3). The performance of the control charts s assessed n terms of the uncondtonal ARL and SDRL. We compute these run length characterstcs n an n-control stuaton and several outof-control stuatons. We consder dfferent shfts n the standard devaton λσ, settng λ equal to.6, 1, 1.2, and 1.4. The performance characterstcs are obtaned by smulaton. The next secton descrbes the smulaton method, followed by the results for the control charts constructed n the uncontamnated stuaton and varous contamnated stuatons. 4.1 Smulaton Procedure For each Phase I dataset of k subgroups of sze n, we determne ˆσ and the control lmts ÛCL and LCL. Let ˆσ be an estmate of λσ based on the th subgroup Y j, j = 1, 2,...,n. Further, let F denote the event that ˆσ s above ÛCL or below LCL. We defne P (F ˆσ ) as the probablty that subgroup generates a sgnal gven ˆσ, that s, (b) P (F ˆσ ) = P ( ˆσ < LCL or ˆσ > ÛCL ˆσ ). Gven ˆσ, the dstrbuton of the run length s geometrc wth parameter P (F ˆσ ). Consequently, the condtonal ARL s gven by 1 E(RL ˆσ ) = P (F ˆσ ). When we take the expectaton over the X j s we get the uncondtonal ARL 1 ARL = E P (F ˆσ ). Ths expectaton s obtaned by smulaton: numerous datasets are generated from the normal dstrbuton or contamnated normal dstrbuton, and for each dataset, E(RL ˆσ ) s computed. By averagng these values, we obtan the uncondtonal value. The uncondtonal standard devaton s

9 8 MARIT SCHOONHOVEN AND RONALD J.M.M. DOES Table 7. ARL and SDRL under normalty for k = 5 ARL SDRL n Chart λ =.6 λ = 1. λ = 1.2 λ = 1.4 λ =.6 λ = 1. λ = 1.2 λ = R s MD s MD R s MD s determned by: MD SDRL = var(rl) = E(var(RL ˆσ )) + var(e(rl ˆσ )) ( ) 1 2 ( ) = 2E E E p(f ˆσ ) p(f ˆσ ) p(f ˆσ ). Enough replcatons of the above procedure were performed to obtan suffcently small relatve estmated standard errors for ARL. The relatve standard error never exceeds.76%. 4.2 Smulaton Results The ARL and SDRL are obtaned n the n-control stuaton (λ = 1) and n the out-of-control stuaton (λ 1). When the process s n control, we want the ARL and SDRL to be close to ther ntended values, namely 37. In the out-of-control Table 8. ARL and SDRL when contamnatons are present n Phase I for k = 5 ARL SDRL n Chart λ =.6 λ = 1. λ = 1.2 λ = 1.4 λ =.6 λ = 1. λ = 1.2 λ = 1.4 N(, 1) & N(, 4) R s (symm) MD s MD R s MD s MD N(, 1) & N(, 4) R s (asymm) MD s MD R s MD s MD

10 A ROBUST STANDARD DEVIATION CONTROL CHART 81 Table 9. ARL and SDRL when contamnatons are present n Phase I for k = 5 ARL SDRL n Chart λ =.6 λ = 1. λ = 1.2 λ = 1.4 λ =.6 λ = 1. λ = 1.2 λ = 1.4 N(, 1) & N(, 4) R s (loc) MD s MD R s MD s MD N(, 1) & N(4, 1) R s MD s MD R s MD s MD stuaton, we want to detect changes n the standard devaton as soon as possble, so the ARL should be as low as possble. The results are obtaned for n = 5, 9 and k = 5, 1. However, as was done for the comparson, we only present the results for k = 5 because the conclusons for k = 1 are not all that dfferent. The tables comparng k = 5 and k = 1 are avalable as Supplementary Materal. Table 7 shows the ARL and SDRL for the stuaton when the Phase I data are uncontamnated and normally dstrbuted. The ARL s very smlar across charts and the SDRL s slghtly hgher for the MD and charts. Tables 8 and 9 show that when there are dsturbances n the Phase I data, the ARL values ncrease (decrease) consderably for λ>1(λ<1) relatve to the normal stuaton. Thus, when the Phase I data are contamnated, changes n the process standard devaton are less lkely to be detected when λ>1, whle there are more sgnals when λ<1. Wth dffuse dsturbances (Table 8 and second half of Table 9), ther mpact s smallest for the charts based on MD,, and. When there are localzed dsturbances (frst half of Table 9), the charts based on R s, MD s, and perform best, because these charts trm extreme subgroups. Note that n a number of cases, the, R s, and MD s charts are ARL-based: the ncontrol ARL s lower than the out-of-control ARL (cf. Jensen et al. 26). Overall, the chart performs best. Under normalty, ths chart almost matches the standard chart based on, and n the presence of any contamnaton, the chart outperforms the alternatves. 5. CONCLUDING REMARKS In ths artcle, we consder several estmators of the standard devaton n Phase I of the control chartng process. We have found that the performance of certan robust estmators s almost dentcal to the pooled subgroup standard devaton under normalty, whle the beneft of usng such robust estmators can be substantal when there are dsturbances. Followng Rocke (1989, 1992), we have consdered estmators that nclude a procedure for subgroup screenng, but whereas Rocke used R, we have used the average devaton from the medan. Ths estmator performs better when there are localzed dsturbances and s much more robust aganst dffuse dsturbances. However, when there are dffuse mean dsturbances, the procedure loses effcency. To address ths problem, we have proposed other algorthms based on a procedure that also screens for ndvdual outlers. The algorthms remove the varaton between subgroups so that only the varaton wthn subgroups s measured. We have shown that these algorthms are very effectve when there are dffuse dsturbances. When there mght also be localzed dsturbances, the method can be combned wth subgroup screenng based on the IQR. The latter procedure reveals a performance very smlar to the robust estmator for the standard devaton control chart

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