UvA-DARE (Digital Academic Repository) A robust Xbar control chart Schoonhoven, M.; Does, R.J.M.M.

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1 UvA-DARE (Digitl Acdemic Repository) A robust Xbr control chrt Schoonhoven, M.; Does, R.J.M.M. Published in: Qulity nd Relibility Engineering Interntionl DOI: 1.12/qre.1447 Link to publiction Cittion for published version (APA): Schoonhoven, M., & Does, R. J. M. M. (213). A robust Xbr control chrt. Qulity nd Relibility Engineering Interntionl, 29(7), DOI: 1.12/qre.1447 Generl rights It is not permitted to downlod or to forwrd/distribute the text or prt of it without the consent of the uthor(s) nd/or copyright holder(s), other thn for strictly personl, individul use, unless the work is under n open content license (like Cretive Commons). Disclimer/Complints regultions If you believe tht digitl publiction of certin mteril infringes ny of your rights or (privcy) interests, plese let the Librry know, stting your resons. In cse of legitimte complint, the Librry will mke the mteril inccessible nd/or remove it from the website. Plese Ask the Librry: or letter to: Librry of the University of Amsterdm, Secretrit, Singel 425, 112 WP Amsterdm, The Netherlnds. You will be contcted s soon s possible. UvA-DARE is service provided by the librry of the University of Amsterdm ( Downlod dte: 3 Nov 218

2 Reserch Article (wileyonlinelibrry.com) DOI: 1.12/qre.1447 Published online 3 October 212 in Wiley Online Librry A Robust X Control Chrt Mrit Schoonhoven* nd Ronld J. M. M. Does** { This rticle studies lterntive stndrd devition estimtors tht serve s bsis to determine the X control chrt limits used for rel-time process monitoring (phse II). Severl existing (robust) estimtion methods re considered. In ddition, we propose new estimtion method bsed on phse I nlysis, tht is, the use of control chrt to identify disturbnces in dt set retrospectively. The method constructs phse I control chrt derived from the trimmed men of the smple interqurtile rnges, which is used to identify out-of-control dt. An efficient estimtor, nmely the men of the smple stndrd devitions, is used to obtin the finl stndrd devition estimte from the remining dt. The estimtion methods re evluted in terms of their men squred errors nd their effects on the performnce of the X phse II control chrt. It is shown tht the newly proposed estimtion method is efficient under normlity nd performs substntilly better thn stndrd methods when disturbnces re present in phse I. Copyright 212 John Wiley & Sons, Ltd. Keywords: ARL; estimtion; phse I; phse II; Shewhrt chrt; stndrd devition; sttisticl process control 1. Introduction The X control chrt is widely pplied technique for effectively monitoring the loction of processes. When the prmeters of qulity chrcteristic of the process re unknown, control chrts cn be pplied in two-stge procedure. In phse I, control chrts re used retrospectively to study historicl dt set nd determine the smples tht re out of control. On the bsis of the resulting reference smple, the process prmeters re estimted nd control limits re clculted for phse II. In phse II, control chrts re used for rel-time process monitoring (cf. Vining 1 ). Recent inititives on phse II control chrts ddress, mongst other things, the joint sttisticl design of the X nd the stndrd devition control chrts (Mukherjee nd Chkrborti 2 nd Chen nd Po 3 ), the vrible smpling intervl X control chrt (Zhng et l. 4 ), the use of different control chrting rules (Kim et l. 5 nd Riz et l. 6 ) nd the use of lterntive estimtors (Schoonhoven et l. 7 nd Schoonhoven nd Does 8 ). Let Y ij, i =1,2,3,...nd j =1,2,..., n denote phse II smples of size n tken in sequence of the process vrible to be monitored. We ssume the Y ij s to be independent nd N(m + ds, s 2 ) distributed, where d is constnt. When d =, the men of the process is in control; otherwise the process men hs chnged. Let Y i ¼ 1 P n n j¼1 Y ij be n estimte of m + ds bsed on the ith smple Y ij, j =1,2,..., n. In prctice, the process prmeters m nd s re usully unknown. Therefore, they must be estimted from smples tken when the process is ssumed to be in control (i.e., phse I). The resulting estimtes re used to monitor the loction of the process in phse II. Define ^m nd ^s s unbised estimtes of m nd s, respectively, bsed on k phse I smples of size n, which re denoted by X ij, i =1, 2,..., k. The control limits cn be estimted by p ^UCL ¼ ^m þ C ffiffiffi p n^s= n ;^LCL ¼ ^m C ffiffiffi n^s= n ; (1) where C n is the fctor such tht the expected probbility of hving flse lrm p equls the desired type I error probbility. Let F i denote the event tht Y i is bove^ucl or below^lcl. Wedefine PF ð i j^m; ^s Þ s the probbility tht smple i genertes signl given ^m nd ^s, tht is, PF i j^m; ^s Þ¼P ^m i <^LCL or ^m i >^UCLj^m; ^s Þ: (2) Given ^m nd ^s, the distribution of the run length (RL) is geometric with prmeter PF ð i j^m; ^s Þ. Consequently, the conditionl verge run length (ARL) is given by 1 EðRLj^m; ^s Þ¼ (3) PF ð i j^m; ^s Þ: Institute for Business nd Industril Sttistics of the University of Amsterdm (IBIS UvA), Plntge Muidergrcht 12, 118 TV, Amsterdm, The Netherlnds *Correspondence to: Mrit Schoonhoven, Institute for Business nd Industril Sttistics of the University of Amsterdm (IBIS UvA), Plntge Muidergrcht 12, 118 TV, Amsterdm, The Netherlnds. E-mil: m.schoonhoven@uv.nl. **Correspondence to: Ronld J. M. M. Does is Professor of Industril Sttistics t the University of Amsterdm, Mnging Director of IBIS UvA, nd Fellow of ASQ. { E-mil: r.j.m.m.does@uv.nl 951

3 In contrst with the conditionl RL distribution, the unconditionl RL distribution tkes into ccount the rndom vribility introduced into the chrting procedure through prmeter estimtion. It cn be obtined by verging the conditionl RL distribution over ll possible vlues of the prmeter estimtes. The unconditionl p is nd the unconditionl ARL is M. SCHOONHOVEN AND R. J. M. M. DOES p ¼ EPF ð ð i j^m; ^s ÞÞ; (4) 1 ARL ¼ Eð Þ: (5) PF ð i j^m; ^s Þ Quesenberry 9 showed for the X control chrt tht the unconditionl ARL is higher thn in the cse where the process prmeters re known. He concluded tht, if limits re to behve like known limits, the number of smples in phse I should be t lest 4/(n 1). Jensen et l. 1 conducted literture survey of the effects of prmeter estimtion on control chrt properties nd identified some issues for future reserch. One of their min recommendtions is to study robust estimtors for m nd s, becuse most studies hve only considered stndrd estimtors. The effect of using these robust estimtors on phse II should lso be ssessed (Jensen et l., 1 p. 36). Schoonhoven et l. 11 nlyzed severl robust loction estimtion methods, including severl methods tht use phse I control chrt. In ddition, they proposed new phse I control chrt derived from the trimen. Their results indicte tht the X phse II control chrt (with s known) bsed on the new estimtion method performs well under normlity nd outperforms the other chrts when contmintions re present in phse I. However, the effect of the process loction method on the performnce of the X phse II control chrt is more limited thn the effect of the stndrd devition estimtor. The present rticle therefore looks t the effect of lterntive stndrd devition estimtors under vrious phse I scenrios. So fr the literture hs proposed severl robust stndrd devition estimtors. Rocke 12 considered the interqurtile rnge nd the 25% trimmed men of the interqurtile rnges. Rocke 13 gve the prcticl detils for the construction of the chrts bsed on these estimtors. Ttum 14 proposed method, constructed round vrint of the biweight A estimtor, tht is resistnt to diffuse disturbnces, tht re, disturbnces tht re eqully likely to perturb ny observtion, nd loclized disturbnces, tht is, disturbnces tht ffect ll observtions in smple. Schoonhoven et l. 15 studied severl estimtors used to construct the stndrd devition phse II control chrt. They found tht Ttum s estimtor is robust ginst diffuse disturbnces but less robust ginst shifts in the process stndrd devition in phse I. They proposed n estimtor bsed on the men devition to the medin supplemented with smple screening in phse I. The dvntge of this estimtor is tht it works well when loclized vrince disturbnces re present but it is less robust when there re symmetric diffuse disturbnces. Finlly, Schoonhoven nd Does 16 proposed stndrd devition estimtion method where the control chrt bsed on the men devition to the medin is supplemented with screening using n individul control chrt in phse I. They investigted the effect of the estimtion method on the stndrd devition control chrt. In this rticle, we develop n estimtion method to derive the stndrd devition for the X control chrt when both m nd s re unknown. Aprt from the new method, severl lterntive estimtion methods re included in the comprison. The methods re evluted in terms of their men squred errors () nd their effect on the X phse II control chrt performnce. We consider the sitution where the phse I dt re uncontminted nd normlly distributed, s well s vrious types of contminted phse I situtions. The pper is structured s follows. Subsequently, we present the estimtion methods for the stndrd devition nd ssess the of the estimtors. In the following sections, we present the design schemes for the X phse II control chrt nd derive the control limits. Next, we describe the simultion procedure nd present the effect of the proposed methods on the phse II performnce. The finl section offers some recommendtions nd issues for future reserch. 2. Proposed phse I estimtors To understnd the behvior of the estimtors, it is useful to distinguish two groups of disturbnces, nmely diffuse nd loclized (cf. Ttum 14 ). Diffuse disturbnces re outliers tht re spred over ll of the smples, wheres loclized disturbnces ffect ll observtions in one smple. We nlyze vrious types of stndrd devition estimtors nd compre them under vrious types of disturbnces. The first nd second subsections introduce the stndrd devition nd loction estimtors, respectively, wheres the second subsection presents the of the stndrd devition estimtors Stndrd devition estimtors Recll tht X ij, i =1,2,..., k nd j =1,2,..., n denote the phse I dt. The X ij s re ssumed to be independent nd N(m, s 2 ) distributed. We denote by X i,(v), v =1,2,..., n the vth order sttistic in smple i. We look t severl robust estimtors proposed in the existing literture nd introduce new method incorporting phse I control chrt. The first estimtor of s is bsed on the men of the smple stndrd devitions 952 where S i is the ith smple stndrd devition defined by S ¼ 1 k X k i¼1 S i (6)

4 S i ¼! 1 X n 1=2 2 X ij X i : n 1 j¼1 An unbised estimtor of s is given by S=c 4 ðnþ, where c 4 (n) isdefined by c 4 ðnþ ¼ 2 1=2 Γðn=2Þ n 1 Γððn 1Þ=2Þ : Note tht this estimtor is slightly less efficient under normlity thn the pooled smple stndrd devition. The ltter, however, is most sensitive to contmintions (Schoonhoven et l. 15 ). The second estimtor is bsed on the men smple rnge where R i is the rnge of the ith smple R ¼ 1 k X k i¼1 R i ¼ X i; ðnþ X i; ð1þ : R i ; (7) An unbised estimtor of s is R=d 2 ðnþ, where d 2 (n) is the expected rnge of rndom N (,1) smple of size n. Vlues of d 2 (n) cn be found in Duncn 17 (Tble M). Rocke 12 proposed the men of the smple interqurtile rnges 1 X k ¼ i ; (8) k where i is the interqurtile rnge of smple i defined by i¼1 i ¼ Q i;3 Q i;1 ; with Q i,q the qth qurtile of smple i (with q = 1, 2, 3). We use the following definitions for the qurtiles: Q i,1 = X i,() nd Q i,3 = X i,(b) with = dn/4e nd b = n + 1, where dze denotes the ceiling function, tht is, the smllest integer not less thn z. This mens tht Q i,1 nd Q i,3 re defined s the second smllest nd the second lrgest observtions, respectively, for 4 n 7 nd s the third smllest nd the third lrgest vlues, respectively, for 8 n 11. An unbised estimtor of s is given by =d, where d is normlizing constnt. The vlue of this normlizing constnt is.99 for n = 5 nd for n =9. Rocke 12 lso proposed the trimmed men of the smple interqurtile rnges 2 3 k 1 ¼ k 2dke X d k 4 e ðþ v 5; (9) v¼dkeþ1 where (v) denotes the vth ordered vlue of the smple interqurtile rnges. We consider the 2% trimmed men of the smple interqurtile rnges, which trims the 1 smllest nd the 1 lrgest smple interqurtile rnges when k = 5 nd the 2 smllest nd the 2 lrgest smple interqurtile rnges when k = 1. An unbised estimtor of s is given by 2 =d I QR 2, where d I QR 2 is normlizing constnt. The vlue of this normlizing constnt is.925 for n = 5 nd 1.18 for n =9. We lso evlute robust estimtor proposed by Ttum. 14 His method hs proven to be robust to both diffuse nd loclized disturbnces. The estimtion method is constructed round vrint of the biweight A estimtor. The method begins by clculting the residuls in ech smple, which involves substrcting the smple medin M i from ech vlue: res ij = X ij M i.ifnis odd, then in ech smple, one of the residuls will be zero nd is dropped. As result, the totl number of residuls is equl to m = nk when n is even nd m =(n 1)k when n is odd. Ttum s estimtor is given by S c ¼ m ðm 1Þ 1=2 P 1=2 4 j: ju ij j<1 res2 ij 1 u 2 ij P ; (1) j: ju ij 1 u j<1 2 ij 1 5u 2 ij P k i¼1 P k i¼1 where u ij = h i res ij /(cm * ), M * is the medin of the bsolute vlues of ll residuls, 8 < 1 E i 4:5; h i ¼ E i 3:5 4:5 < E i 7:5; : c E i > 7:5; nd E i = i /M *. The constnt c is tuning constnt. Ech vlue of c leds to different estimtor. Ttum showed tht c = 7 gives n estimtor tht loses some efficiency when no disturbnces re present but gins efficiency when disturbnces re present. We pply 953

5 this vlue of c in our study. Note tht we hve h i = E i 3.5 for 4.5 < E i 7.5 in the equtions insted of h i = E i 4.5 s presented by Ttum (Ttum, 14 p.129). An unbised estimtor of s is given by S c =d ðc; n; kþ, where d * (c, n, k) is the normlizing constnt. In our study, we use the corrected normlizing constnts given in Schoonhoven et l. 15 We now present new estimtion method bsed on the principle of phse I control chrting (cf. Jones-Frmer et l. 18 ). We build phse I control chrt using robust estimtor for the stndrd devition, nmely 2. A disdvntge of this estimtor is tht it is not very efficient under normlity. To ddress this, we use 2 to construct the phse I limits with which we screen the estimtion dt for disturbnces, but then use the efficient estimtor S to obtin stndrd devition estimte from the remining dt. The phse I stndrd devition control chrt limits re given by^ucl I QR 2 ¼ U n 2 =d I QR 2 nd^lcl I QR 2 ¼ L n 2 =d I QR 2. For simplicity, we derive U n nd L n from the nd.135 quntiles of the distribution of /d. These quntiles re obtined by simultion, nd 1,, simultion runs re used. The respective vlues of U n nd L n re 3.22 nd.35 for n = 5 nd nd.145 for n =9. We then plot the i /d s of the phse I smples on the phse I control chrt. Chrting the insted of the smple stndrd devition or the smple rnge ensures tht loclized vrince disturbnces re identified nd smples tht contin only one single outlier re retined. A stndrd devition estimte tht is expected to be robust ginst loclized vrince disturbnces is bsed on the men of the smple interqurtile rnges of the smples tht fll between the control limits ¼ 1 X k i 1^LCL I QR 2 i=d ^UCL ð 2 i Þ; iek M. SCHOONHOVEN AND R. J. M. M. DOES with K the set of smples which re not excluded nd k the number of non-excluded smples. The resulting estimte =d is unbised. Although the remining phse I smples re expected to be free from loclized vrince disturbnces, they could still contin diffuse disturbnces. To eliminte such disturbnces, the next step is to screen the individul observtions using phse I individuls control chrt. To screen the individul observtions, we determine the residuls in ech smple by subtrcting the trimen vlue from ech observtion in the corresponding smple: resid ij = X ij TM i with TM i ¼ Q i;1 þ 2Q i;2 þ Q i;3 =4: Note tht Q i,2 is the medin of smple i. Substrcting the smple trimens ensures tht the vribility is mesured within smples nd not between smples. According to Tukey, 19 using the trimen insted of the men or the medin gives more useful ssessment of loction or centering. The control limits of the individuls chrt re given by^ucl I QR ¼ 3 =d nd^lcl I QR ¼ 3 =d. The residuls resid ij tht fll bove^ucl I QR or below^lcl I QR re considered out of control nd their corresponding observtions re removed from the phse I dt set. The finl estimte is the men of the smple stndrd devitions S i nd is clculted from the observtions deemed to be in control S ¼ 1 X S k i X ij 1^LCLI iek QR residij ^UCL X ij ; (11) I QR with K the set of smples which re not excluded nd k the number of non-excluded smples. The normlizing constnt is.98 for n = 5 nd.984 for n = 9. This dptively trimmed stndrd devition is denoted by. The proposed stndrd devition estimtors re summrized in Tble I Loction estimtor The forementioned stndrd devition estimtors re used to construct the X phse II control limits. To ensure fir comprison, we use the sme loction estimtor in ech cse. The loction estimtion method uses procedure similr to. This procedure ws proposed by Schoonhoven et l. 11 nd turned out to perform much better thn the stndrd estimting procedures bsed on, for exmple, the men, medin, trimmed men nd Hodges Lehmnn estimtor. The procedure consists of two steps. In the first step, we determine loction estimte tht is robust ginst both loclized nd diffuse men disturbnces, nmely the 2% trimmed men of the smple trimens Tble I. Proposed stndrd devition estimtors Estimtors Nottion 954 Men of smple stndrd devitions S Men of smple rnges R Men of smple interqurtile rnges 2% trimmed men of smple interqurtile rnges 2 Ttum s estimtor control chrt with screening

6 1 TM ¼ k 2 k 2 k d e X d k 4 e v¼dkeþ1 3 TM ðþ v 5: Note tht we strt with the entire dt set. The respective upper nd lower control limits for the smple loction re given by ^UCL TM 2 ¼ TM p 2 þ 3^s= ffiffiffi n nd^lcltm 2 ¼ TM p 2 3^s= ffiffiffi n, where s is estimted by the corresponding stndrd devition estimtor from Tble I. We then plot the TM i s of the phse I smples on the control chrt. Chrting the TM i s insted of the X i s ensures tht loclized disturbnces re identified nd smples tht contin only one single outlier re retined. A loction estimte tht is expected to be robust ginst loclized men disturbnces is the men of the smple trimens of the smples tht fll between the control limits TM ¼ 1 X k TM i 1^LCL TM2 TM i ^UCL iek TM 2 ðtm i Þ; with K * the set of smples which re not excluded nd k * the number of non-excluded smples. Although the remining phse I smples re expected to be free from loclized men disturbnces, they could still contin diffuse disturbnces. To eliminte such disturbnces, the next step is to screen the individul observtions using phse I individuls control chrt with respective upper nd lower control limits given by^ucltm ¼ TM þ 3^s nd^lcltm corresponding stndrd devition estimtor from Tble I. The observtions X ij tht fll bove^ucltm ¼ TM 3^s, where s is estimted by the or below^lcl TM re considered out of control nd removed from the phse I dt set. The finl estimte is the men of the smple mens nd is clculted from the observtions deemed to be in control X ¼ 1 k X iek 1 n i X jen i X ij 1^LCL TM X ij ^UCL X TM ij ; (12) with K the smples which re not excluded, k the number of non-excluded smples, N i the observtions tht re not excluded in smple i nd n i the number of non-excluded observtions in smple i Efficiency of proposed stndrd devition estimtors For comprison purposes, we ssess the of the proposed stndrd devition estimtors s ws performed in Ttum. 14 The will be estimted s ¼ 1 N X N i¼1 ^s i 2; s where ^s i is the vlue of the unbised estimte in the ith simultion run nd N is the number of simultion runs. We consider the uncontminted cse, tht is, the sitution where ll X ij re from the N (,1) distribution s well s four types of disturbnces (cf. Ttum 14 ): 1. A model for diffuse symmetric vrince disturbnces in which ech observtion hs 95% probbility of being drwn from the N (,1) distribution nd 5% probbility of being drwn from the N(, ) distribution, with = 1.5, 2.,..., 5.5, A model for diffuse symmetric vrince disturbnces in which ech observtion is drwn from the N(, 1) distribution nd hs 5% probbility of hving multiple of w 2 1 vrible dded to it, with the multiplier equl to.5, 1.,..., 4.5, A model for loclized vrince disturbnces in which observtions in five (when k = 5) or 1 (when k = 1) smples re drwn from the N(, ) distribution, with = 1.5, 2.,..., 5.5, A model for diffuse men disturbnces in which ech observtion hs 95% probbility of being drwn from the N(, 1) distribution nd 5% probbility of being drwn from the N(b, 1) distribution, with b =.5, 1.,..., 9., 9.5. The is obtined for k = 5, 1 smples of sizes n = 5, 9. The number of simultion runs N is equl to 5,. Figures 1 4 show the of the proposed estimtors. The following results cn be observed. The stndrd estimtors S nd R re not robust ginst either loclized or diffuse disturbnces. The is less efficient under normlity when there re no contmintions, but performs resonbly well when there re diffuse disturbnces. The reson why performs so well in these situtions is tht it trims the highest nd lowest observtions in ech smple. However, this estimtor remins bised when there re symmetric diffuse disturbnces becuse the trimming method does not tke the distribution of the contmintions into ccount. Furthermore, this estimtor is not efficient when there re loclized vrince disturbnces s it trims only the observtions within the smple insted of the smple interqurtile rnges. An estimtor tht combines within-smple nd between-smple trimmings, nmely 2, performs resonbly well for ll types of contmintions. However, its efficiency is reltively low under normlity. isefficient under normlity s well s for contminted dt but reltively less so when the contmintion consists of loclized vrince disturbnces. 955

7 () (c) (b) (d) Figure 1. Men squred errors of estimtors when symmetric diffuse vrince disturbnces re present: () n =5,k =5; (b) n =5,k =1; (c) n =9,k =5; nd (d) n =9,k = () (c) (b) (d) 956 Figure 2. Men squred errors of estimtors when symmetric diffuse vrince disturbnces re present: () n =5,k =5; (b) n =5,k = 1; (c) n =9,k =5; nd (d) n =9,k = 1

8 () (c) (b) (d) Figure 3. Men squred errors of estimtors when loclized vrince disturbnces re present: () n =5,k = 5; (b) n =5,k = 1; (c) n =9,k = 5; nd (d) n =9,k = b () b (b) b (c) b Figure 4. Men squred errors of estimtors when diffuse men disturbnces re present: () n =5,k = 5; (b) n =5,k = 1; (c) n =9,k = 5; nd (d) n =9,k = 1 (d) 957

9 The estimtor is slightly less efficient under normlity thn the stndrd estimtors, but much more robust thn nd 2. Moreover, it shows outstnding performnce when contmintions re present. We cn therefore conclude tht this estimtor effectively filters out extreme observtions. 3. Derivtion of the phse II control limits We now turn to the effect of the proposed estimtors on the performnce of the X phse II control chrt. The formule for the X control limits with estimted limits re given by (1). The fctor C n tht is used to obtin ccurte control limits when the process prmeters re estimted is derived, such tht the probbility of flse signl equls the chosen type I error probbility p. The fctors cnnot be obtined esily in nlytic form. Therefore, they re obtined by mens of simultion. The chosen type I error probbility p is.27. 5, simultion runs re used. The resulting fctors re presented in Tble II. 4. Control chrt performnce In this section, we evlute the effect on X phse II performnce of the proposed stndrd devition estimtors. We consider the sme phse I situtions s those used to ssess the with, b nd the multiplier equl to 4 to simulte the contminted cse (cf. the Section on Efficiency of Proposed Stndrd Devition Estimtors). The performnce of the phse II control chrts is ssessed in terms of the unconditionl p nd ARL s well s the conditionl ARL. The conditionl ARL vlues express the ARL vlues for the control limits ssocited with the 2.5% nd 95.7% quntiles of simulted p in the in-control sitution. We consider different shifts of size ds in the men in phse II, nmely d equl to,.25,.5 nd 1. The performnce chrcteristics re obtined by simultion. The next section describes the simultion method, followed by the results for the control chrts constructed in the uncontminted sitution nd vrious contminted situtions Simultion procedure The performnce chrcteristics p nd ARL for estimted control limits re determined by verging the conditionl chrcteristics, tht is, the chrcteristics for given set of estimted control limits, over ll possible vlues of the control limits. Recll the definitions 1 of pf ð i j^m; ^s Þfrom (2), ERL^m; ð j ^s Þfrom (3), p ¼ EpF ð ð i j^m; ^s ÞÞfrom (4) nd ARL ¼ E pðf i j^m;^s Þ from (5). These expecttions will be obtined by simultion: numerous dt sets re generted, nd for ech dtset, pf ð i j^m; ^s Þ nd ERL^m; ð j ^s Þ re computed. By verging these vlues we obtin the unconditionl vlues. Enough replictions of the forementioned procedure were performed to obtin sufficiently smll reltive estimted stndrd errors for p nd ARL. The reltive estimted stndrd error is the estimted stndrd error of the estimte reltive to the estimte. The reltive stndrd error of the estimtes is never higher thn.8% Simultion results The performnce metrics re obtined in the in-control sitution (d = ) s well s in the out-of-control sitution (d 6¼ ). When d =, the process is in control, so we wnt p to be s low s possible nd ARL to be s high s possible. When d 6¼, tht is, in the out-of-control sitution, we wnt to chieve the opposite. Tble III shows the performnce metrics for the X phse II chrts under normlity. In this cse, we hve estimted both the in-control m nd s in phse I. Compred with the X phse II performnce presented in Schoonhoven et l., 11 where only the men ws estimted to isolte the effect of estimting the loction prmeter, the ARL vlues re much higher thn the desired 37. Thus, Tble II. Fctors C n to determine Phse II control limits Fctors for control limits Chrt n =5 n =9 k =5 k = 1 k =5 k = S R

10 Tble III. Unconditionl p nd ARL nd (in prentheses) the upper nd lower conditionl ARL vlues under normlity p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d =1 5 5 S (155; 1256) (64.1; 418) (18.6; 85.7) (3.21; 7.94) R (163; 1318) (63.3; 489) (18.4; 98.1) (3.19; 8.58) (11; 3127) (75.7; 1838) (22.6; 37) (3.48; 17.4) (96.4; 589) (42.8; 1332) (13.5; 227) (2.69; 14.9) (145; 1374) (81.2; 493) (23.1; 98.6) (3.56; 8.63) (135; 1536) (57.8; 451) (17.1; 91.2) (3.6; 8.32) 1 S (194; 818) (77.1; 398) (21.5; 83.7) (3.48; 7.66) R (193; 857) (76.7; 282) (21.4; 61.8) (3.48; 6.5) (146; 1487) (61.1; 522) (17.9; 13) (3.14; 8.89) (133; 1929) (56.6; 553) (16.8; 18) (3.3; 9.26) (189; 877) (81.6; 29) (22.6; 63.2) (3.57; 6.59) (179; 94) (8.; 296) (22.3; 64.3) (3.53; 6.68) 9 5 S (189; 846) (148; 157) (23.8; 24.6) (2.31; 2.48) R (186; 923) (53.1; 171) (1.9; 26.2) (1.75; 2.54) (142; 1512) (38.2; 257) (8.61; 35.8) (1.61; 2.94) (135; 1858) (36.7; 388) (8.37; 48.7) (1.6; 3.36) (182; 885) (65.8; 184) (12.7; 27.6) (1.85; 2.59) (177; 931) (127; 171) (21.1; 26.3) (2.21; 2.55) 1 S (232; 643) (56.4; 132) (11.4; 21.6) (1.79; 2.33) R (223; 668) (61.2; 135) (12.1; 21.9) (1.83; 2.35) (Continues) 959

11 96 Tble III. Continued. p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d = (182; 941) (47.2; 226) (1.; 32.1) (1.71; 2.75) (174; 166) (46.6; 193) (9.94; 28.7) (1.7; 2.65) (225; 67) (61.1; 13) (12.1; 21.4) (1.83; 2.33) (219; 681) (75.8; 143) (14.1; 22.7) (1.92; 2.38)

12 Tble IV. Unconditionl p nd ARL nd (in prentheses) the upper nd lower conditionl ARL vlues when symmetric vrince disturbnces re present in phse I p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d =1 5 5 S (546; ) (343; ) (76.1; 1675) (7.12; 58.7) R (562; ) (275; ) (61.7; 1758) (6.35; 61.1) (166; 9948) (19; 2979) (3.1; 449) (4.1; 33.3) (144; ) (7.4; 27) (2.3; 413) (3.34; 22.4) (221; 382) (12; 129) (27.1; 29) (3.94; 14.) (17; 364) (69.9; 86) (19.9; 157) (3.33; 11.7) 1 S (88; ) (39; 511) (66.9; 713) (6.8; 32.5) R (912; ) (32; 6155) (65.4; 837) (6.73; 36.1) (24; 3632) (99.8; 952) (26.6; 171) (3.91; 12.4) (24; 462) (83.7; 146) (23.; 185) (3.61; 13.1) (33; 2126) (126; 785) (32.1; 146) (4.36; 11.) (235; 1696) (19; 57) (49.3; 11) (5.4; 8.83) 9 5 S (934; ) (185; ) (67.7; 732) (2.6; 14.) R (1162; ) (23; ) (32.7; 1246) (2.8; 19.3) (218; 3166) (62.1; 459) (12.2; 56.3) (1.83; 3.66) (196; 369) (19; 533) (18.5; 63.2) (2.11; 3.87) (271; 1945) (85.7; 348) (15.4; 45.) (2.; 3.26) (225; 1618) (64.3; 274) (12.5; 37.6) (1.85; 3.) 1 S (1446; ) (244; 238) (34.4; 23) (2.89; 7.15) R (1851; ) (35; 5238) (4.9; 39) (3.13; 1.2) (Continues) 961

13 962 Tble IV. Continued. p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d = (287; 1831) (67.6; 416) (13.1; 51.2) (1.89; 3.42) (263; 1953) (61.9; 317) (12.2; 42.1) (1.84; 3.17) (354; 1367) (91.8; 24) (16.3; 33.9) (2.6; 2.86) (287; 116) (71.4; 195) (13.6; 29.) (1.92; 2.67)

14 Tble V. Unconditionl p nd ARL nd (in prentheses) the upper nd lower conditionl ARL vlues when symmetric vrince disturbnces re present in phse I p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d =1 5 5 S (549; ) (277; ) (62.3; ) (6.38; 7196) R (575; ) (194; ) (95.4; ) (5.4; 1172) (158; 1324) (72.3; 2951) (2.6; 446) (3.38; 23.6) (132; ) (62.8; 8338) (18.4; 1152) (3.17; 42.) (189; 2565) (76.4; 72) (21.4; 132) (3.47; 1.5) (155; 2292) (77.7; 647) (22.; 123) (3.49; 1.1) 1 S (117; ) (467; ) (94.7; ) (8.34; 42) R (197; ) (42; ) (83.1; ) (7.75; 26) (223; 4523) (19; 1148) (29.; 2) (4.7; 13.8) (188; 351) (8.2; 16) (22.3; 179) (3.54; 12.7) (254; 1525) (99.4; 45) (26.4; 91.1) (3.91; 8.3) (29; 1312) (83.8; 397) (23.; 82.) (3.62; 7.77) 9 5 S (1173; ) (28; ) (3.5; ) (2.72; 166) R (148; ) (26; ) (36.; ) (2.92; 314) (194; 2921) (49.7; 444) (1.4; 54.7) (1.73; 4.6) (177; 384) (61.5; 768) (12.1; 82.3) (1.81; 4.29) (232; 1423) (131; 243) (21.3; 34.4) (2.23; 2.88) (198; 1248) (63.; 216) (12.3; 31.3) (1.83; 2.76) 1 S (2429; ) (577; ) (66.; 4594) (3.86; 45.8) R (3124; ) (532; ) (62.6; ) (3.82; 77.3) (Continues) 963

15 964 Tble V. Continued. p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d = (254; 1592) (6.7; 443) (12.1; 52.6) (1.83; 3.44) (235; 1674) (58.4; 354) (11.7; 45.4) (1.81; 3.24) (293; 126) (123; 192) (2.1; 28.6) (2.21; 2.64) (251; 894) (81.4; 178) (14.9; 27.) (1.97; 2.57)

16 Tble VI. Unconditionl p nd ARL nd (in prentheses) the upper nd lower conditionl ARL vlues when loclized vrince disturbnces re present in phse I p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d =1 5 5 S (1843; ) (154; ) (192; 2442) (12.8; 76.9) R (185; ) (767; ) (144; 266) (1.8; 8.7) (918; ) (743; ) (15; ) (1.8; 231) (193; ) (76.8; 4467) (41.4; 636) (3.48; 3.) (224; 3717) (9.4; 1536) (24.5; 257) (3.74; 15.8) (155; 2899) (216; 78) (7.2; 144) (6.5; 11.2) 1 S (2865; ) (78; 9525) (144; 1222) (11.1; 47.3) R (287; ) (134; ) (184; 1456) (12.8; 53.) (163; ) (264; ) (144; 322) (1.8; 9.) (285; 6672) (17; 1654) (28.1; 272) (4.6; 16.9) (39; 2132) (114; 629) (29.6; 12) (4.18; 9.89) (218; 163) (89.9; 468) (24.4; 94.4) (3.73; 8.47) 9 5 S (2724; ) (55; 6959) (59.9; 487) (3.72; 11.4) R (268; ) (773; ) (82.4; 93) (4.26; 16.) (1671; ) (277; ) (38.; 899) (3.2; 16.7) (229; 4581) (87.2; 621) (15.6; 71.4) (2.; 4.12) (273; 213) (118; 348) (19.6; 45.1) (2.18; 3.26) (176; 1211) (96.9; 225) (17.; 32.2) (2.4; 2.79) 1 S (3975; ) (69; 2965) (56.6; 249) (4.21; 8.) R (3776; ) (1534; 3841) (142; 33) (5.53; 8.84) (Continues) 965

17 966 Tble VI. Continued. p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d = (2599; ) (44; 6258) (5.8; 453) (3.47; 11.2) (31; 2447) ( ) (45.6; 49.1) (2.2; 3.41) (359; 1422) (79.6; 249) (14.8; 34.9) (1.99; 2.9) (226; 842) (57.4; 184) (11.6; 27.5) (1.8; 2.58)

18 Tble VII. Unconditionl p nd ARL nd (in prentheses) the upper nd lower conditionl ARL vlues when diffuse men disturbnces re present in phse I p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d =1 5 5 S (991; ) (341; ) (72.5; 1754) (7.15; 6.) R (13; ) (168; ) (216; 4737) (13.4; 115) (245; ) (18; 6419) (28.4; 866) (4.5; 36.4) (176; ) (83.2; 9626) (23.1; 1253) (3.6; 45.4) (299; 9368) (146; 3666) (36.7; 359) (4.67; 25.9) (175; 5726) (18; 1457) (29.6; 244) (4.7; 15.7) 1 S (1554; ) (823; ) (155; 2638) (11.2; 75.) R (1651; ) (951; ) (176; 1756) (12.1; 57.6) (366; 819) (142; 1924) (35.4; 31) (4.63; 18.4) (262; 5767) (99.9; 157) (26.5; 26) (3.93; 16.2) (433; 48) (172; 2467) (421.3; 389) (5.7; 2.5) (247; 2762) (94.9; 1218) (25.4; 213) (3.83; 13.9) 9 5 S (158; ) (2351; 5964) (216; 425) (6.5; 1.4) R (1923; ) (512; ) (6.; 3614) (3.67; 33.9) (278; 5152) (64.8; 693) (12.7; 77.7) (1.87; 4.29) (235; 4557) (391; 964) (56.5; 98.8) (3.29; 4.74) (361; 3751) (1; 533) (17.4; 63.2) (2.11; 3.87) (228; 2339) (159; 496) (24.8; 58.8) (2.37; 3.66) 1 S (221; ) (45; ) (54.6; 893) (3.55; 14.9) R (2834; ) (999; 7126) (11; 488) (4.67; 11.1) (Continues) 967

19 968 Tble VII. Continued. p ARL n k Chrt d = d =.25 d =.5 d =1 d = d =.25 d =.5 d = (376; 2822) (82.2; 592) (15.1; 67.5) (2.1; 3.92) (316; 2457) (71.6; 373) (13.6; 47.9) (1.92; 3.38) (484; 2483) (11; 521) (17.6; 61.1) (2.14; 3.74) (33; 1479) (93.3; 259) (16.4; 36.) (2.5; 2.94)

20 lso estimting the process stndrd devition hs substntilly more impct on the X phse II control chrt thn only estimting the process men. The conditionl ARL vlues re presented in prentheses. The first vlue in prentheses represents the ARL for the control limits ssocited with the 97.5% quntile of the simulted p in the in-control sitution, wheres the second vlue represents the ARL for the control limit ssocited with the 2.5% quntile of the simulted p in the in-control sitution. The results show tht the conditionl ARL vlues vry quite strongly, even when k equls 1. In the bsence of ny contmintion, the chrts bsed on S, R, nd show comprble performnce. The chrts bsed on nd 2 re less powerful under normlity. The nlysis shows tht, when there re disturbnces in the phse I dt, the performnce of ll chrts chnges considerbly: p decreses nd the ARL vlues increse. Thus, when the phse I dt re contminted, shifts in the process men re less quickly detected. When symmetric disturbnces re present (Tble IV), their impct is the smllest for the chrts bsed on, 2, nd. These chrts re lso lest ffected when there re symmetric disturbnces (Tble V). Both tbles show tht the chrt bsed on outperforms the others. When there re loclized disturbnces (Tble VI), the chrts bsed on the estimtors nd perform best, the reson being tht these chrts trim extreme smples. Finlly, in the cse of diffuse men disturbnces (Tble VII), the chrts bsed on 2, nd perform better thn the other chrts. Overll, the chrt performs best. Under normlity, the chrt essentilly mtches the performnce of the stndrd chrts bsed on S nd R nd, in the presence of ny contmintion, the chrt outperforms the lterntives. 5. Concluding remrks nd future reserch In this rticle, we hve considered severl estimtion methods for the stndrd devition prmeter. The of the estimtors hs been ssessed under vrious circumstnces: the uncontminted sitution nd vrious situtions contminted with diffuse symmetric nd symmetric vrince disturbnces, loclized vrince disturbnces nd diffuse men disturbnces. Moreover, we hve investigted the effect of estimting the stndrd devition estimtor on the X phse II control chrt performnce when the methods re used to determine the phse II limits. The stndrd methods suffer from number of problems. Estimtors tht re bsed on the principle of trimming observtions (e.g., ) perform resonbly well when there re diffuse disturbnces but not when there re loclized disturbnces. In the ltter sitution, estimtors tht include method to trim smple sttistics (e.g., 2 ) re efficient. All of these methods re bised when there re symmetric disturbnces, s the trimming principle does not tke into ccount the symmetry of the disturbnce. A phse I nlysis using control chrt to study historicl dtset retrospectively nd trim the dt dptively does tke into ccount the distribution of the dt nd is therefore very suitble for use during the estimtion of s. In this rticle, we hve proposed new type of phse I nlysis. The initil estimte of s for the phse I control chrt is given by n estimtor tht is robust ginst both diffuse nd loclized disturbnces, nmely 2. We hve shown tht this estimtor is not very efficient under normlity. However, when 2 is only used to construct the phse I control chrt limits, nd when the stndrd estimtion method S is used to determine the finl estimte of s fter screening, the resulting estimtor () isefficient under normlity. Moreover, outperforms the other estimtion methods when there re contmintions. It is therefore suitble method for determining the vlue of s in the X phse II control chrt limits. References 1. Vining G. Technicl dvice: phse I nd phse II control chrts. Qulity Engineering 29; 21: Mukherjee A, Chkrborti S. A distribution-free control chrt for the joint monitoring for loction nd scle. Qulity nd Relibility Engineering Interntionl 212; 28: Chen H, Po Y. The joint economicl sttisticl design of X nd R chrts for nonnorml dt. Qulity nd Relibility Engineering Interntionl 211; 27: Zhng Y, Cstgliol P, Wu Z, Khooe MBC. The vrible smpling intervl X chrt with estimted prmeters. Qulity nd Relibility Engineering Interntionl 212; 28: Kim YB, Hong JS, Lie CH. Economic-sttisticl design of 2-of-2 nd 2-of-3 runs rule scheme. Qulity nd Relibility Engineering Interntionl 29; 25: Riz M, Mehmood R, Does RJMM. On the performnce of different control chrting rules. Qulity nd Relibility Engineering Interntionl 211; 27: Schoonhoven M, Riz M, Does RJMM. Design schemes for the X control chrt. Qulity nd Relibility Engineering Interntionl 29; 25: Schoonhoven M, Does RJMM. The X control chrt under non-normlity. Qulity nd Relibility Engineering Interntionl 21; 26: Quesenberry CP. The effect of smple size on estimted limits for X nd X control chrt. Journl of Qulity Technology 1993; 25: Jensen WA, Jones-Frmer LA, Chmp CW, Woodll WH. Effects of prmeter estimtion on control chrt properties: literture review. Journl of Qulity Technology 26; 38: Schoonhoven M, Nzir ZN, Riz M, Does RJMM. Robust loction estimtors for the X control chrt. Journl of Qulity Technology 211; 43: Rocke DM. Robust Control Chrts. Technometrics 1989; 31: Rocke DM. X Q nd R Q chrts: robust control chrts. The Sttisticin 1992; 41: Ttum LG. Robust estimtion of the process stndrd devition for control chrts. Technometrics 1997; 39:

21 15. Schoonhoven M, Riz M, Does RJMM. Design nd nlysis of the stndrd devition control chrt with estimted prmeters. Journl of Qulity Technology 211; 43: Schoonhoven M, Does RJMM. A robust stndrd devition control chrt. Technometrics 212; 54: Duncn AJ. Qulity Control nd Industril Sttistics. R.D. Irwin Inc.: Homewood, 1986; Jones-Frmer LA, Jordn V, Chmp CW. Distribution-free phse I control chrts for subgroup loction. Journl of Qulity Technology 29; 41: Tukey JW. Explortory Dt Anlysis. Addison-Wesley: Boston, Authors' biogrphies Dr. M. Schoonhoven is Senior Consultnt of Sttistics t IBIS UvA. Her current reserch focuses on the design of control chrts for nonstndrd situtions. Dr. Ronld J. M. M. Does is Professor of Industril Sttistics t the University of Amsterdm, Mnging Director of IBIS UvA, nd Fellow of ASQ. 97