Chapter 3: Signed-rank charts

Transcription

1 Chaer : gned-ran chars.. The hewhar-ye conrol char... Inroducon As menoned n Chaer, samles of fxed sze are aen a regular nervals and he long sasc s hen loed. The queson s: Whch qualy arameer should be used as he long sasc? In Chaer he sgn es sasc N was descrbed and was menoned ha he sgn es sasc s only nfluenced by he sgns of he devaons ( x j θ. There s an alernave sasc ha can be used o rac he locaon of a rocess. The sasc s a funcon of boh he magnudes and sgns of he x θ s, called he sgned-ran sasc. ( j... Defnon of he sgned-ran es sasc The sgned-ran es s a nonaramerc es ha can be used o es hyoheses on or consruc confdence nervals (see Gbbons and Charabor ( for he medan of any symmerc connuous oulaon dsrbuon. Le X,...,, X X n denoe he h (,,... samle or subgrou of ndeenden observaons of sze n > from a rocess wh an unnown connuous dsrbuon funcon denoed by F. Le θ denoe he nown n-conrol locaon arameer (also called he arge value. Le devaons, j θ R j denoe he ran of he absolue x, whn he subgrou ( θ, x θ x θ x for,,....,..., n Then R j s referred o as he whn-grou absolue ran of he devaons. The sgned-ran es sasc s gven by R n j sgn( x θ R for,,... (. j j where sgn (x -,, f x <,, >.

2 ... long sasc The sgned-ran es sasc, R (gven n (., s used as he long sasc on he hewhar sgned-ran char. If he long sasc R falls beween he wo conrol lms, ha s, LCL < R < UCL, he rocess s consdered o be n-conrol. If he long sasc R falls on or ousde one of he conrol lms, ha s he rocess s consdered o be ou-of-conrol. R LCL or R UCL, sasc The long sasc s lnearly relaed o he well-nown Wlcoxon sgned-ran T n hrough he formula (see Bar (, age 44, equaon.4 n( n R Tn (. where T n n j ψ ( x θ R, ψ ( x, f x, >. j j Examle. A wo-sded hewhar sgned-ran char for he Mongomery ( son rng daa We llusrae he hewhar-ye sgned-ran char usng he same se of daa from Mongomery ( ha was used n examle.. We assume ha he underlyng dsrbuon s symmerc wh a nown medan θ 74 mm. anel a of Table. exhbs he ndvdual observaons of 5 ndeenden samles, each of sze 5.e. n 5. The absolue devaons x j θ and sgn ( x j θ are shown n anel b and anel c of Table., resecvely. The nown arge value s aen o be 74, ha s, θ 74. The whn-grou absolue ran of he devaons R j and he sgn x j θ R values are shown n anel a and anel b of Table., ( j resecvely. anel c of Table. holds he sgned-rans.e. R for,,,..., 5.

3 Table.. Daa and calculaons for he sgned-ran char *. anel a anel b anel c amle number x x x x 4 x * ee A rogram 5 n Aendx B for he calculaon of he values n Table..

4 Table.. Calculaons for he sgned-ran char *. anel a anel b anel c amle number R R R R 4 R 5 R Le ARL and FAR denoe he n-conrol average run lengh and he false alarm rae for he uer one-sded hewhar sgned-ran conrol char, resecvely. For an uer onesded char we would ae UCL 5 snce s relaed o a false alarm rae of. (. FAR and an n-conrol average run lengh of ( ARL - see Table.. Alhough he n-conrol average run lengh of s far from he desred value, whch s generally aen o be 7 or 5, s he bes under resen condons. The false alarm rae ( FAR and he n-conrol average run lengh ( ARL for he symmerc wo-sded hewhar sgned-ran char can be obaned hrough he relaonshs FAR FAR and ARL ARL, resecvely (see Bar (. A symmerc wo-sded char s obaned by choosng LCL UCL. We ae UCL 5 for he wo-sded hewhar sgned-ran char, * ee A rogram 5 n Aendx B for he calculaon of he values n Table.. 4

5 snce s relaed o a false alarm rae of.66 ( FAR FAR.. 66 and ARL an n-conrol average run lengh of ( ARL. The wo-sded sgned-ran char s shown n Fgure. wh UCL 5, CL and LCL 5. Fgure.. gned-ran conrol char for Mongomery ( son rng daa. The char sgnals a samle number. Therefore, a search for assgnable causes s necessary. I aears mos lely ha he rocess medan has shfed uwards from he arge value of 74mm...4. Deermnaon of char consans The conrol lms n examle. were chosen o gve a ceran false alarm rae or nconrol ARL. Values of varous conrol lms are gven by Bar (. Bar ncluded he followng able n hs arcle whch gves he false alarm raes and he n-conrol average run lenghs for he uer one-sded hewhar sgned-ran chars based on subgrous of szes n 4, 5 and 6. 5

6 Table.. FAR s and ARL s for he uer one-sded hewhar sgned-ran char. UCL n 4 n 5 n 6 ARL FAR ARL FAR ARL FAR Table. shows he false alarm raes and he n-conrol average run lenghs for he uer one-sded hewhar sgned-ran char as calculaed usng he null dsrbuon of he Wlcoxon sgned-ran sasc (see Hollander and Wolfe (97 and Bar (. In Table. we see ha here are some dulcaes n he daa. We consder a secfc examle o shed lgh on he occurrence of hese dulcaes. uose n 5 and UCL. Then FAR ( R In - conrol (.5 (usng (.. The las robably T n equals ( 4. 6, because T n T has zero robably a.5. When n 5 and n UCL we have ha FAR ( R In - conrol ( 4. 6 (by usng T n he null dsrbuon of he Wlcoxon sgned-ran sasc. nce FAR ( 4.6 for wo dfferen values of he uer conrol lm, we have T n dulcaes n he daa. Ths examle ons ou an error * n Table of Bar (. The robably of (.5 equals ( 4 whch equals.6 (and no.98 T n T n corresondng o ( as reored by Bar s ( aer. Ths ye of correcon T n was aled o he oher enres of Bar s ( Table and are gven n Table. of hs hess. The false alarm raes and n-conrol average run lenghs for he wo-sded hewhar sgned-ran char were calculaed usng A (wh he arorae correcons made and are shown n Table.4. * Ths error s also oned ou by Charabor and Erylmaz (7.

7 Table.4. FAR s and ARL s for he wo-sded hewhar sgned-ran char *. UCL n 4 n 5 n 6 n 7 n 8 n 9 n ARL FAR ARL FAR ARL FAR ARL FAR ARL FAR ARL FAR ARL FAR * ee A rogram 6 n Aendx B for he calculaon of he values n Table.4. Ths able s an exenson of Tables and gven n Bar (. 7

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9 ..5. ummary The sgned-ran es s a oular nonaramerc es for he medan of a symmerc connuous oulaon. The sgned-ran es s more owerful han he sgn es, bu whle he sgn es s alcable for all connuous dsrbuons, he assumon of symmery mus be made, n addon, for he sgned-ran es. Furhermore, he sgn es ales o all ercenles, whereas he sgned-ran es s roosed only for he medan. Anoher drawbac of he sgned-ran char s ha he FAR values for he char are oo hgh (n oher words he ARL values are oo shor unless he subgrou sze s large. One way o remedy hs roblem s o use some sgnalng rules o enhance he sensvy of he chars. Ths wll be consdered nex... The hewhar-ye conrol char wh runs-ye sgnalng rules... Inroducon In addon o defnng warnng lms or zones on conrol chars (see econ., we can exend he exsng chars by ncororang varous sgnalng rules nvolvng runs of he long sasc. The sgnalng rules consdered nclude he followng: A rocess s declared o be ou-of-conrol when (a a sngle on (charng sasc los ousde he conrol lm(s (-of- rule (b consecuve ons (charng sascs lo ousde he conrol lm(s (of- rule or (c exacly of he las w ons (charng sascs lo ousde he conrol lm(s (-of-w rule. We can consder hese sgnalng rules where boh and w are osve negers wh w and w. Rule (a s he smles and s he mos frequenly used n he leraure. Thus, he -of- rule corresonds o he usual conrol char, where a sgnal s gven when a long sasc falls ousde he conrol lm(s. Rules (a and (b are secal cases of rule (c; rules (b and (c have been used n he conex of sulemenng he hewhar chars wh warnng lms and zones. Rules (a, (b and (c have been suded by varous auhors (see for examle Klen ( and Khoo (4. Klen ( suggesed wo rules namely he -of- and -of- rules. Boh conrol chars are easly mlemened and have beer ARL erformance han he -of- rule. Khoo (4 conduced a sudy of he ARL erformance of he -of-, -of-, -of-4, -of- and -of-4 chars and concluded ha he -of- 4 char s he mos sensve scheme for deecng small rocess shfs. 9

10 Charabor and Erylmaz (7 consdered smle alernaves o he Bar (4 s class of nonaramerc chars, usng he sgned-ran sasc bu ncororang runs rules of he ye dscussed above o defne new sgnalng rules. If we se equal o n rule (b above, we oban he smles of he -of- ye rules whch are called he -of- DR and he -of- KL chars. The -of- KL char sgnals, for examle, when he wo mos recen sgnedran sascs boh fall eher on or above or on or below he conrol lms. The -of- DR char s almos smlar, bu here a sgnal s ndcaed when boh of he sgned-ran sascs fall eher boh on or above or boh on or below or one on or above (below and he nex one on or below (above he conrol lms. I s shown ha he new chars are nonaramerc, have much smaller FAR (and hus larger ARL han he -of- sgned-ran char of Bar. Moreover, he new chars have beer ou-of-conrol erformance han he -of- sgned-ran char for heavy-aled and sewed dsrbuons such as he Cauchy. We llusrae hese rocedures usng he Mongomery ( son rng daa.... Examle Examle. A wo-sded hewhar sgned-ran char wh sgnalng rules for he Mongomery ( son rng daa We llusrae he sgned-ran char wh sgnalng rules usng he Mongomery ( son rng daa. Recall ha he daase conans 5 samles (each of sze 5. The sgned-ran sascs were calculaed and gven n Table. and grahcally reresened n Fgure.. The symmerc wo-sded conrol lms for he -of- and -of- sgned-ran chars are gven by Charabor and Erylmaz (7 for n 4,5, 6 and. The able for samles of sze 5 s gven here for reference.

11 Table.5. False alarm raes and n-conrol ARL values for he wo-sded -of- and -of- sgned-ran chars under DR and KL schemes, n 5 *. -of- -of- DR -of- KL UCL ARL FAR ARL DR, FAR DR, ARL KL, FAR KL, For n 5, he conrol lms for he -of- (Bar s char, he -of- DR and he -of- KL chars, based on he sgned-ran sasc, are se a ± 5. These yeld FAR values.66,.9, and.9, resecvely. If he conrol lms were aen o be ±, he FAR would have been much hgher:.5,.56, and.78, resecvely. Alhough he conrol lms are he same, namely ± 5, he sgnalng rules are que dfferen oeraonally and he erformance of he resulng chars urn ou o be que dfferen. The -of- char sgnals when he frs sgned-ran sasc falls on or ousde of eher of he wo conrol lms; he - of- KL char sgnals when, for he frs me, wo consecuve sgned-ran sascs fall eher on or above or on or below he wo conrol lms, whle he -of- DR char sgnals when for he frs me wo consecuve sgned-ran sascs fall on or ousde he conrol lms, eher boh on or above, or boh on or below, or one on or above he nex on or below, or one on or below and he nex on or above. On he erformance sde, noe ha he -of- R char has a FAR of.66 and an ARL of aroxmaely. Thus many more false alarms wll be sgnaled by hs char leadng o a ossble loss of me and resources. Comared o ha, he -of- KL char has a FAR of.9 and an ARL of 56.4, whereas he -of- DR char has a FAR of.9 and an ARL of 7.5. Thus boh of hese run-rule-enhanced chars rovde reasonable and raccal false alarm raes and can be used n racce, deendng on he ye of shf one execs. From Fgure. we see ha he DR and KL -of- sgned-ran chars boh sgnal a samle, ndcang a mos lely uward shf n he rocess medan. The -of- sgnedran char, on he oher hand, sgnals earler, a samle, bu noe he much hgher FAR of.66 (and corresondngly a much lower and less desrable ARL, 5.97 assocaed wh hs char. I s neresng o noe ha, as shown n Mongomery (, for hese daa he * Table.5 aears n Charabor and Erylmaz (7, Table.

12 hewhar X char ndcaes a shf n he mean a samle for hese daa. However, he ey dfference s ha an alcaon of he hewhar char can rase several quesons such as he form of he underlyng dsrbuon (small n 5, and more moranly abou he n-conrol (sable erformance of he char n erms of he FAR (or he ARL, snce s nown ha he n-conrol erformance of he hewhar X char s no robus n ycal qualy conrol alcaons. Comared o hs, he roosed nonaramerc chars rovde a more generally alcable alernave monorng scheme wh a nown (sable/robus n-conrol erformance and a beer or equal ou-of-conrol erformance han he -of- sgned-ran char.... ummary In hs secon we examned sgned-ran conrol chars wh runs-ye sgnalng rules. Human, Charabor and m (8 recenly suded hewhar-ye sgn chars wh runsye sgnalng rules. These chars are smlar n sr o he hewhar-ye sgned-ran chars wh runs-ye sgnalng rules (see econ.. In he aer by Human e al. hey derved exressons for he run lengh dsrbuons usng Marov chan heory. The n-conrol and ou-of-conrol erformance of hese chars were suded and comared o hose of he exsng sgned-raned chars under he normal, double exonenal and Cauchy dsrbuons, usng he ARL, DRL, FAR and some ercenles of he run lengh. These runs rules enhanced sgn chars have he advanage ha one does no have o assume symmery of he underlyng dsrbuon and hey can be aled n suaons where he daa are dchoomous.

13 .. The abular CUUM conrol char... Inroducon Bar and Reynolds (979 nvesgaed he CUUM char usng he Wlcoxon sgnedran sasc. They used mehods ha are analogous o he mehods used on he CUUM sgn char (see econ., ha s, a Marov chan aroach s used o fnd he momens and oher characerscs of he run lengh dsrbuon for he CUUM sgned-ran char.... One-sded conrol chars... Uer one-sded conrol chars Fu, rng and Xe ( and Fu and Lou ( resened hree resuls ha mus be sasfed before mlemenng he fne-sae Marov chan aroach. Le be a fne-sae homogenous Marov chan on he sae sace Ω wh a ranson robably marx (TM such ha ( Ω { ς, ς,..., ς r s} where ς < ς <... < ς r s h and ς rs s an absorben sae; ( he TM s gven by TM ] for,,..., r s and j,,..., r s where r [ j denoes he number of non-absorben * saes and s he number of absorben saes, resecvely, and ( he sarng value should equal zero wh robably one, ha s, ( (hs s o ensure ha he rocess sars n-conrol. Assume ha he Marov chan (, ( and (, hen he formulas gven n (.4 o (.45 hold. sasfes condons The me ha he rocedure sgnals s he frs me such ha he fne-sae Marov chan eners he sae ς rs where he sae sace s gven by Ω { ς, ς,..., }, and { h, max{, R } mn (. ς r s * The ransen (non-absorben saes are he saes for whch evenual reurn s unceran. If a sae s enered once and s never lef, he sae s sad o be absorben.

14 .4. The EWMA conrol char.4.. Inroducon In hs secon, he aroach aen by Lucas and accucc (99 s exended o he use of he sgned-ran sasc resulng n an EWMA sgned-ran char ha accumulaes he sascs R, R, R,.... econ.4 s analogous o econ.4 where he aroach aen by Lucas and accucc (99 was exended o he use of he sgn sasc resulng n an EWMA sgn char. Therefore, he reader s frequenly referred bac o econ.4 hroughou hs secon..4.. The roosed EWMA sgned-ran char A nonaramerc EWMA-ye of conrol char based on he sgned-ran sasc (recall ha R n j sgn ( x θ R can be obaned by relacng X n exresson (.5 of econ j j.4 wh R. The EWMA sgned-ran char accumulaes he sascs R, R, R,... wh he long sascs defned as Z R ( Z (. where < s a consan called he weghng consan. The sarng value Z could be aen o equal zero,.e. Z. The EWMA sgned-ran char s consruced by long (or me. If he long sasc Z falls beween he wo conrol lms, ha s, he rocess s consdered o be n-conrol. If he long sasc conrol lms, ha s Z agans he samle number LCL < Z < UCL, Z falls on or ousde one of he Z LCL or Z UCL, he rocess s consdered o be ou-of-conrol. The exac conrol lms and he cener lne of he EWMA sgned-ran conrol char can be obaned by relacng σ and θ by.4 o oban σ R and, resecvely, n exresson (.55 of econ 88

15 UCL Lσ CL R LCL Lσ R ( ( ( (. (. mlarly, he seady-sae conrol lms can be obaned by relacng σ and θ by and, resecvely, n exresson (.56 o oban σ R UCL Lσ R LCL Lσ R (. where σ R denoes he n-conrol sandard devaon of he sgned-ran sasc R f here are no es whn a subgrou. The n-conrol sandard devaon of R s gven by σ var( R n( n vart R and ha n( n ( 6 n T (recall ha. Ths s obaned by usng he relaonsh beween n( n R T f here are no es whn a subgrou and he fac n( n (n var( T (see Gbbons and Charabor ( age R.4.. Marov-chan aroach Lucas and accucc (99 evaluaed he roeres of he connuous sae Marov chan by dscrezng he nfne sae TM. Ths rocedure enals dvdng he nerval beween he UCL and he LCL no N subnervals of wdh δ. Then he long sasc, Z, s sad o be n he non-absorbng sae j a me f j δ < Z j δ where j denoes he mdon of he h j nerval. Z s sad o be n he absorbng sae f Z falls on or ousde one of he conrol 89

16 9 lms, ha s, LCL Z or UCL Z. Le j denoe he robably of movng from sae o sae j n one se,.e. ( sae n j sae o Movng j. To aroxmae hs robably we assume ha he long sasc s equal o whenever s n sae. For all j non-absorbng we oban ( j j j Z Z < δ δ. By usng he defnon of he long sasc gven n exresson (. we oban ( < < δ δ δ δ j j j j j R R ( ( ( ( ( recall ha ( n n T R < δ δ j j j n n T ( ( ( ( ( < ( ( ( ( ( ( n n T n n j j δ δ. (. For all j absorbng we oban ( ( ( ( j UCL n n T LCL n n T UCL R LCL R UCL R LCL R Z UCL Z Z LCL Z ( ( ( ( ( ( ( ( ( ( ( ( n n UCL T n n LCL T. (.4 nce he values LCL, UCL, δ,, n, and j are nown consans he Wlcoxon sgned-ran robables n exressons (. and (.4 can easly be calculaed. The robables for he Wlcoxon sgned-ran sascs are gven n Table H of Lehmann (975 for

17 samles szes u o and hey are abulaed (more recenly n Table H of Gbbons and Charabor ( for samle szes u o 5. Once he one-se ranson robables are calculaed, he TM can be consruced and Q s gven by TM [ j ] (wren n aroned form where he essenal ranson ' robably sub-marx Q s he marx ha conans all he ranson robables of gong from a non-absorbng sae o a non-absorbng sae, Q : ( NA NA, conans all he ranson robables of gong from each non-absorbng sae o he absorbng saes, : ( NA A, ' ( conans all he ranson robables of gong from each absorbng sae o he non-absorbng saes. ' s a row vecor wh all s elemens equal o zero, because s mossble o go from an absorbng sae o a non-absorbng sae, because once an absorbng sae s enered, s never lef, ' : ( A NA, and reresens he scalar value one. The robably of gong of gong from an absorbng sae o an absorbng sae s equal o one, because once an absorbng sae s enered, s never lef, : ( A A. The one-se TM s used o calculae he execed value (ARL, he second raw momen, he varance, he sandard devaon and he robably mass funcon (mf of he run-lengh varable N whch are gven n equaons (.4 o (.45. Examle. The EWMA sgned-ran char where he samle sze s even ( n 6 The EWMA sgned-ran char s nvesgaed for a smoohng consan of. (. and a muller of ( L. The seady-sae conrol lms are gven by UCL Lσ R LCL Lσ R 9

18 n( n (n where L,., and σ R 9. 59, snce σ R 6 6(6 ( Clearly, we only have o calculae he UCL snce LCL UCL. We oban. UCL Therefore, LCL Ths Marov-chan rocedure enals dvdng he nerval beween he UCL and he LCL no N subnervals of wdh δ. For hs examle N s aen o equal 4. Fgure. llusraes he aronng of he nerval beween he UCL and he LCL no subnervals. Fgure.. aronng of he nerval beween he UCL and he LCL no 4 subnervals. gven by From Fgure. we see ha here are 4 non-absorbng saes,.e. r 4. The TM s 9

19 4 4 Q4 4 4 TM 4. 4 ' Table.. Calculaon of he one-se robables n he frs row of he TM. ( Movng o sae n ( < Z δ Z sae δ from (. ( δ ( n( n ( δ ( n( n < T wh δ. 64,., L and (4.755 < T.9 ( T ( T 4 57 from Gbbons and Charabor ( 64 ( Movng o sae n sae ( δ < Z δ Z from (. (.658 < T ( T δ ( n( n < T ( δ ( n( n ( 9

20 ( Movng o sae n ( < Z δ Z sae δ from (. ( 8.7 < T δ ( n( n < T.658 ( δ ( n( n ( ( Movng o sae n ( < Z δ Z sae δ from (. δ ( n( n < T ( < T 8.7 ( δ ( n( n ( ( Movng o sae 4 n ( Z LCL Z ( Z UCL Z 4 sae from (.4 LCL ( n( n UCL ( n( n T T T ( ( T.9 The one-se robables n he remanng rows can be calculaed smlarly. Therefore, Q4 4 4 he TM s gven by TM ' 4 Oher values of he muller (L and he smoohng consan ( were also consdered and he resuls are gven n Tables. and.. 94

21 Table.. The n-conrol average run lengh ( ARL, sandard devaon of he run lengh ( DRL, h h h h 5, 5, 5, 75 and h 95 ercenle values * for he EWMA sgned-ran char when n 6 and N 5,.e. here are 5 subnervals beween he lower and uer conrol lm..5.. L L L ** (,, 6, 5, 5 (, 5, 9, 8, (,, 4,, (,,, 7, (, 4,, 48, (, 6,, 5, (, 6,, 47, (, 87, 4, 5, ** The nverse of he marx ( I Q does no exs and as a resul he ARL (gven by ( N ( I Q E ξ can no be calculaed for hs combnaon of (, L. In examle. we consdered a samle sze ha may be consdered small. The resuls are gven for a larger samle sze ( n for varous values of and L n Table.. Table.. The n-conrol average run lengh ( ARL, sandard devaon of he run lengh ( DRL, h h h h 5, 5, 5, 75 and h 95 ercenle values for he EWMA sgned-ran char when n and N 5,.e. here are 5 subnervals beween he lower and uer conrol lm..5.. L L L (,, 6,, 9 (, 7, 8, 98, (,, 4, 9, (,,, 7, (, 6, 9, 7, (, 6,, 48, (,, 956, 5, (, 76,, 5, (, 8, 6, 474, 5 * The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 8 n Aendx B for he calculaon of he values n Table.. The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 8 n Aendx B for he calculaon of he values n Table.. 95

22 These ables can be exended by changng he samle sze (n, he number of subnervals beween he lower and uer conrol lm (N, he muller (L and he smoohng consan ( n A rogram 8 for he EWMA sgned-ran char gven n Aendx B. From Tables. and. we see ha he ARL, DRL and ercenles ncrease as he value of he muller (L ncreases. From Table. we fnd an n-conrol average run lengh of 6.4 for n when he muller s aen o equal ( L and he smoohng consan. (.. The char erformance s good, snce he aaned n-conrol average run lengh of 6.4 s n he regon of he desred n-conrol average run lengh whch s generally aen o be 7 or ummary The EWMA conrol char s one of several charng mehods amed a correcng a defcency of he hewhar char - nsensvy o small shfs. Lucas and accucc (99 have nvesgaed some roeres of he EWMA char under he assumon of ndeenden normally dsrbued observaons, whereas n hs secon we have descrbed and evaluaed he nonaramerc EWMA sgned-ran char. The man advanage of he nonaramerc EWMA char s ha here s no need o assume a arcular aramerc dsrbuon for he underlyng rocess (see econ.4 for oher advanages of he nonaramerc EWMA char. 96

23 where h s he decson nerval and s he reference value (see econ.. for a dealed dscusson on how he values of and h are chosen. Equaon (. s obaned by relacng wh R n (.46. N The dsrbuon of sgned-ran sasc R can easly be obaned from he dsrbuon of he Wlcoxon T (recall ha R n( n T. The robables for he Wlcoxon sgned-ran sascs are gven n Table H of Lehmann (975 for samles szes u o and hey are abulaed (more recenly n Table H of Gbbons and Charabor ( for samle szes u o 5. Examle. An uer one-sded CUUM sgned-ran char where he samle sze s even (n4 The sascal roeres of an uer one-sded CUUM sgned-ran char wh a decson nerval of 6 ( h 6, a reference value of ( and a samle sze of 4 ( 4 We sar by examnng he mf of he well-nown Wlcoxon sgned-ran sasc long sasc R s lnearly relaed o T. n s examned. T, snce he 4

24 Table.6. Enumeraon for he dsrbuon of Value of T Rans assocaed wh osve dfferences T for a samle sze of 4. Number of samle ons u ( ( T ( T {,}; {} 5 4 {,}; {4} 7 5 {,4}; {,} 9 6 {,,}; {,4} 7 {,,4}; {,4} 8 {,,4} 4 9 {,,4} 5 {,,,4} From Table.6 f follows ha he mf of T ( f T (,,,8,9,, 4,5,6,7 oherwse T when he samle sze s 4 s ( n n The values of and n( n 4(4 R are eher he even or he odd negers beween (and ncludng n( n, deendng on wheher n( n s even or odd. In examle. whch s even and as a resul he ossble values for R are even negers beween - and nclusve. Thus, we have ha R. In boh cases (wheher n( n s even or odd he sum ( R wll be an neger snce boh R and are negers. For hs examle, he reference value s aen o be equal o wo, because hs leads o he sum ( R beng equal o even values whch reduces he sze of he sae sace for he Marov chan. For h 6 we have ha Ω ς, ς, ς, ς } {,,4,6} wh { 5

25 ς < ς < ς < ς h. The sae sace s calculaed usng equaon (. and he calculaons are shown n Table.7. Table.7. Calculaon of he sae sace when h 6, and n 4. max { } { { R } R R, R mn h,max, - - * Table.8. Classfcaon of he saes. ae number Descron of he sae Absorben (A/ Non-absorben (NA NA NA 4 NA 6 A From Table.8 we see ha here are hree non-absorben saes,.e. r, and one absorben sae,.e. s. Therefore, he corresondng TM wll be a ( r s ( r s 4 4 marx. I can be shown (see Table.9 ha he TM s gven by TM Q ' * Noe: nce only he sae sace needs o be descrbed, whou loss of generaly,. Any oher ossble value for can be any value from would lead o he same Ω and we herefore ae, Ω. 6

26 where he essenal ranson robably sub-marx Q : ( NA NA s an r r marx, : ( NA A s an ( r s column vecor, ' : ( A NA s a ( r s row vecor and : ( A A reresens he scalar value one. by The one-se ranson robables are calculaed by subsung n( n T and subsung n values for h,, and. The calculaon of he onese ranson robables are gven for llusraon n Table.9. R n exresson (. j Ω The robables n he las column of he TM can be calculaed usng he fac ha (see equaon (.8. Therefore, j ; 6 ( 4 ( 9 6 ( 4 ( ; 7 46 ( ( ; ( (

27 Table.9. The calculaon of he ranson robables when h 6, and n 4. ( ( mn { 6, max{, R } ( max {, R } ( R ( R ( T ( T 6 ( mn 6, max, R ( { { } ( max {, R } ( R ( T ( T ( 4 ( mn { 6, max{,4 R } ( max {, R } ( R ( R ( T ( T * ( 6 ( ( mn { 6, max{, R } ( max {, R } ( R ( R 4 ( T 4 ( T 7 ( ( mn { 6, max{, R } ( max {, R } ( R ( T ( T 6 4 ( 4 ( mn { 6, max{,4 R } ( max {, R } ( R ( R ( T ( T 5 6 ( 6 4 ( 4 ( mn { 6, max{, R } 4 ( max {, R } 4 ( R 4 ( R 6 ( T 6 ( T 8 4 ( 4 ( mn { 6, max{, R } 4 ( max {, R } 4 ( R 4 ( T 4 ( T 7 44 ( 4 4 ( mn { 6, max{,4 R } 4 ( max {, R } 4 ( R 4 ( R ( T ( T 6 64 ( 4 6 Usng he TM he ARL can be calculaed usng ( I Q ARL ξ. A well-nown concern s ha moran nformaon abou he erformance of a conrol char can be mssed when only examnng he ARL (hs s esecally rue when he rocess dsrbuon s sewed. Varous auhors, see for examle, Radson and Boyd (5 and Charabor (7, have * The robably equals zero, because s mossble o go from an absorben sae o a non-absorben sae. 8

28 suggesed ha one should examne a number of ercenles, ncludng he medan, o ge he comlee nformaon abou he erformance of a conrol char. Therefore, we now also consder ercenles. The ρ h ercenle s defned as he smalles neger l such ha he cdf s a leas ( ρ%. Thus, he ρ h ercenle l s found from ( N l ρ. The medan ( 5 ercenle wll be consdered, snce s a more reresenave erformance measure han he h ARL. The frs and hrd quarles ( 5 and h 75 ercenles wll also be consdered, snce conans he mddle half of he dsrbuon. The als of he dsrbuon should also be examned and herefore he h 5 and ercenles s shown below for llusraon uroses. h 95 ercenles are calculaed. The calculaon of hese h Table.. Calculaon of he ercenles when h 6, and n 4 N ( N l The 5 h, 5 h, 5 h, 75 h and 95 h ercenles.5 ρ.5 (smalles neger such ha he cdf s a leas.5.54 ρ.5 (smalles neger such ha he cdf s a leas ρ.5 5 (smalles neger such ha he cdf s a leas ρ.75 9 (smalles neger such ha he cdf s a leas ρ.95 9 (smalles neger such ha he cdf s a leas *. * ee A rogram 7 n Aendx B for he calculaon of he values n Table.. The value of he run lengh varable s only shown u o N for llusraon uroses. 9

29 The formulas of he momens and some characerscs of he run lengh dsrbuon have been suded by Fu, rng and Xe ( and Fu and Lou ( see equaons (.4 o (.45. By subsung ξ (, equaons, we oban he followng: Q 9 and 7 no hese ARL E ξ ( N ( I Q 6. 8 ( N ( I Q( I Q E ξ ( N ( E( 6. DRL Var( N E N h 5 ercenle ρ 5 h 5 ercenle ρ 5 Medan h 5 ercenle ρ 5 5 h 75 ercenle ρ 75 9 h 95 ercenle ρ 95 9 Oher values of h, and n were also consdered and he resuls are gven n Table.. 4

30 Table.. The n-conrol average run lengh ( ( DRL, h h h h 5, 5, 5, 75 and h 95 ercenle values * for he uer one-sded CUUM sgnedran char when n ARL, sandard devaon of he run lengh h (,,,, (,,, 4, (,, 4, 7, (,, 6,, (, 5,,, (,,, 4, (,, 4, 7, (,, 6,, (, 5,,, (,,, 6, (,, 5, 9, (, 4, 9, 8, (,, 4, 7, (, 4, 7, 4, (,, 5,, 9 o sasfy In order o allow for he ossbly of song afer one grou, he values of h s aen n( n h. For examle, for 4 n( n 4(4 be smaller han or equal o, snce. n and, he reference value h s aen o The fve ercenles are dslayed n boxlo-le grahs n Fgure. for all he ( h, - combnaons ha are shaded n Table.. I clearly shows he effecs of h and on he run lengh dsrbuon. Fgure. descrbes he run-lengh dsrbuon when he rocess s nconrol. We would refer a boxlo wh a hgh valued (large n-conrol average run lengh and a small sread. The boxlos are classfed no caegores, namely, small ( h 4, * The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 7 n Aendx B for he calculaon of he values n Table.. I should be noed ha hese boxlo-le grahs dffer from sandard box los. In he laer case he whsers are drawn from he ends of he box o he smalles and larges values nsde secfed lms, whereas, n he case of he boxlo-le grahs, he whsers are drawn from he ends of he box o he 5 h and 95 h ercenles, resecvely. In hs hess boxlo wll refer o a boxlo-le grah from hs on forward. 4

31 moderae ( 5 h 8 and large ( h 9. If he sum of he reference value,, and he decson nerval, h, s small (moderae or large, he corresondng boxlo s classfed under small (moderae or large. For examle, where h 4, he boxlo s classfed as small, snce he ARL, DRL and ercenle values are small for n 4. In conras, where h, he boxlo s classfed as large, snce he n 4. ARL, DRL and ercenle values are large for (4, (, (h, 5 5 (4, (, (4, 4 (, 6 (4, 6 (, 8 'mall' 'Moderae' 'Large' (h, Fgure.. Boxlo-le grahs for he n-conrol run lengh dsrbuon of varous uer onesded CUUM sgned-ran chars when n 4. The whsers exend o he 5 h and he 95 h ercenles. The symbols, and denoe he ARL, DRL * and MRL, resecvely. * For ease of nerreaon, he sandard devaon (as measure of sread s ncluded n he (locaon measures of ercenles. 4

32 Examle.4 An uer one-sded CUUM sgned-ran char where he samle sze s odd (n5 The sascal roeres of an uer one-sded CUUM sgned-ran char wh a decson nerval of 6 ( h 6, a reference value of ( and a samle sze of 5 ( 5 We sar by examnng he mf of he well-nown Wlcoxon sgned-ran sasc long sasc R s lnearly relaed o T (see equaon (.. n s examned. T, snce he Table.. Enumeraon for he dsrbuon of Value of T Rans assocaed wh osve dfferences Number of samle ons u ( T for a samle sze of 5. ( T ( T {,}; {} 5 4 {,}; {4} 7 5 {,4}; {,}; {5} 6 {,,}; {,5}; {,4} 7 {,,4}; {,5}; {,4} 8 {,,5}; {,,4}; {,5} 9 9 {,,5}; {,,4}; {4,5} {,,,4}; {,4,5}; {,,5} 5 {,,,5}; {,4,5} 7 {,,4,5}; {,4,5} 9 {,,4,5} 4 {,,4,5} 5 {,,,4,5} 4

33 f T From Table. f follows ha he mf of ( T (,,,,4,5, 4,, 5, 6, 7,8,9, oherwse T when he samle sze s 5 s The reference value was aen o be equal o hree, because hs leads o he sum ( R beng equal o even values whch reduces he sze of he sae sace for he Marov chan. For h 6 we have ha Ω ς, ς, ς, ς } {,,4,6} wh ς < ς < ς < ς h. { The sae sace s calculaed usng equaon (. and he calculaons are shown n Table.. Table.. Calculaon of he sae sace when h 6, and n 5. max { } { { R } R R, R mn h,max, -5-8 * * Noe: nce only he sae sace needs o be descrbed, whou loss of generaly,. Any oher ossble value for can be any value from would lead o he same Ω and we herefore ae, Ω. 44

34 Table.4. Classfcaon of he saes. ae number Descron of he sae Absorben (A/ Non-absorben (NA NA NA 4 NA 6 A From Table.4 we see ha here are hree non-absorben saes,.e. r, and one absorben sae,.e. s. Therefore, he corresondng TM wll be a ( r s ( r s 4 4 marx. I can be shown (see Table.5 ha he TM s gven by TM Q ' where he essenal ranson robably sub-marx Q : ( NA NA s an r r marx, : ( NA A s an ( r s column vecor, ' : ( A NA s a ( r s row vecor and : ( A A reresens he scalar value one. The calculaon of he one-se ranson robables are gven for llusraon n Table.5. Recall ha he robables n he las column of he TM are calculaed usng he fac ha j (see equaon (.8. Therefore, j Ω ; 6 ( 4 ( 9 6 ( 4 ( 5 ; 46 ( ( 7 ; ( (

35 Table.5. The calculaon of he ranson robables when h 6, and n 5. ( ( mn { 6, max{, R } ( max {, R } ( R ( R ( T 5 ( T 9 ( mn 6, max, R ( { { } ( max {, R } ( R ( R 9 ( T 5 ( T 8 4 ( 4 ( mn { 6, max{,4 R } ( max {, R } ( R ( R ( T 5 ( T 7 6 * ( 6 ( ( mn { 6, max{, R } ( max {, R } ( R ( R 5 ( T 5 5 ( T ( ( mn { 6, max{, R } ( max {, R } ( R ( R ( T 5 ( T 9 4 ( 4 ( mn { 6, max{,4 R } ( max {, R } ( R ( R ( T 5 ( T 8 6 ( 6 ( 4 ( mn { 6,max{, R } 4 ( max {, R } 4 ( R 4 ( R 7 ( T 5 7 ( T 4 ( 4 ( mn { 6,max{, R } 4 ( max {, R } 4 ( R 4 ( R 5 ( T 5 5 ( T 44 ( 4 4 ( mn { 6,max{,4 R } 4 ( max {, R } 4 ( R 4 ( R ( T 5 ( T 9 64 ( 4 6 * The robably equals zero, because s mossble o go from an absorben sae o a non-absorben sae. 46

36 The formulas of he momens and some characerscs of he run lengh dsrbuon have been suded by Fu, rng and Xe ( and Fu and Lou ( see equaons (.4 o (.45. By subsung ξ (, equaons, we oban he followng: Q 9 and no hese ARL E ξ ( N ( I Q ( N ( I Q( I Q 6. 4 E ξ ( N ( E( 5. DRL Var( N E N h 5 ercenle 5 h 5 ercenle 5 Medan h 5 ercenle 5 4 h 75 ercenle 75 8 h 95 ercenle 95 Oher values of h, and n were also consdered and he resuls are gven n Table.. 47

37 Table.. The n-conrol average run lengh ( ARL, sandard devaon of he run lengh ( DRL, ercenle values * for he uer one-sded CUUM sgned-ran char when n (,,,, h (,,, 4, 8 (,,, 5, (,, 4, 7, 4 (,, 5, 9, 8 (,, 6,, (,,, 4, (,,, 6, (,, 5, 9, (,, 8, 5, (, 5,,, (,,, 44, (,,, 6, (,, 4, 8, (,, 7, 4, (, 5,,, (, 9,, 4, (,, 4, 8, (,, 7,, (, 5,,, (, 9,, 4, (,, 6,,.4. (, 4, 9, 8, (, 8, 8, 5, (, 4, 8, 4,. 9. (, 6, 4, 8, (, 5,,, 4 h h h h 5, 5, 5, 75 and (, 4, 8, 4, 8 h 95 * The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 7 n Aendx B for he calculaon of he values n Table.. 48

38 The fve ercenles are dslayed n boxlo-le grahs n Fgure. for all he ( h, - combnaons ha are shaded n Table.. I clearly shows he effecs of h and on he run lengh dsrbuon. Fgure. descrbes he run-lengh dsrbuon when he rocess s nconrol. We would refer a boxlo wh a hgh valued (large n-conrol average run lengh and a small sread. The boxlos are classfed no caegores, namely small ( h 5, moderae ( 6 h and large ( h (4, (, (h, 4 (4, (, (6, (4, 5 (4, (, 'mall' 'Moderae' 'Large' (h, Fgure.. Boxlo-le grahs for he n-conrol run lengh dsrbuon of varous uer onesded CUUM sgned-ran chars when n 5. The whsers exend o he 5 h and he 95 h ercenles. The symbols, and denoe he ARL, DRL * and MRL, resecvely. * For ease of nerreaon, he sandard devaon (as measure of sread s ncluded n he (locaon measures of ercenles. 49

39 Examles. and.4 llusraed he Marov chan aroach used o calculae run lengh characerscs for n even and odd, resecvely. On he erformance sde, noe ha he larges n-conrol average run lengh ha he uer one-sded CUUM sgned-ran can oban s Therefore, for a samle sze of 4 he larges n. ARL equals 4 (hs s obaned when h and 8. Thus, a large number of false alarms wll be sgnaled by hs char leadng o a ossble loss of me and resources. Comared o hs, for a samle of sze 5 he larges ARL equals 5 (hs s obaned when h and. Boh examles consdered samle szes ha may be consdered small. ome resuls wll be gven for larger samle szes ( n 6 and. 5

40 Table.7. The n-conrol average run lengh ( ARL, sandard devaon of he run lengh ( DRL, ercenle values * for he uer one-sded CUUM sgned-ran char when n h h h h 5, 5, 5, 75 and h (,,,, (,,, 4, (,,, 5, (,,, 6, (,, 5, 9, (,, 6,, (, 4, 9, 8, 7..8 (, 6, 5, 9, 6..5 (,,, 44, (4, 9, 45, 89, (,,, 4, (,,, 5, (,,, 6, (,, 4, 8, (,, 6,, (, 4, 9, 7, (, 6, 5, 9, (, 9,, 44, (4, 8, 44, 87, (,,, 4, (,,, 6, (,, 4, 8, (,, 6,, 4..6 (, 4, 9, 7, (, 6, 4, 8, (, 9,, 4, (4, 8, 4, 84, (,,, 5, (,, 4, 7, (,, 6,,.6.6 (, 4, 8, 5, (, 6,, 5, (, 9,, 4, (, 7, 9, 78, (,, 4, 7, (,, 5, 9, (,, 7, 4, (, 5,,, (, 8, 8, 6, (, 5, 5, 69, (,, 5, 8, (,, 6,, (, 5,, 8, (, 7, 5,, (,, 9, 57, (,, 5,, (, 4, 8, 5, (, 6,, 4, (,,, 44, (,, 6,,.4. (, 5,, 8, (, 8, 7,, 67 h (, 4, 8, 4, (, 6,,, 47 * The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 7 n Aendx B for he calculaon of he values n Table.7. 5

41 (4, (, 5 (h, (4, (, 5 (4, (, (, (8, 'mall' 'Moderae' 'Large' (h, Fgure.4. Boxlo-le grahs for he n-conrol run lengh dsrbuon of varous uer onesded CUUM sgned-ran chars when n 6. The whsers exend o he 5 h and he 95 h ercenles. The symbols, and denoe he ARL, DRL * and MRL, resecvely. * For ease of nerreaon, he sandard devaon (as measure of sread s ncluded n he (locaon measures of ercenles. The boxlos are classfed no caegores, namely small ( h 7, moderae ( 8 h and large ( h 7. 5

42 Table.8. The n-conrol average run lengh ( ARL, sandard devaon of he run lengh ( DRL, h h h h 5, 5, 5, 75 and ercenle values * for samles of sze n for h,4,..., 4 and,,..., for he uer one-sded CUUM sgned-ran char. h h (,,,, 5 (,,,, 6 (,,,, 7 (,,, 4, 7 (,,, 4, 8 (,,, 4, 9 (,,, 5,.6.8 (,,,, (,,,, (,,, 4, (,,, 4, (,,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5,, (,, 6,, (,,,, (,,, 4, (,,, 4, (,,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5,, (,, 6,, (,, 7, 4,.84.7 (,,, 4, 7..6 (,,, 4, (,,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5,, (,, 6,, (,, 7, 4,.4.8 (, 4, 9, 7, (,,, 4, (,,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5,, (,, 6,, (,, 7, 4, (, 4, 9, 7, (, 5,,, (,,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5, 9, (,, 6,, (,, 7,, (, 4, 8,, (, 5,,, (, 6,, 5, (,,, 5, (,,, 6, (,, 4, 7, (,, 4, 8, (,, 5, 9, (,, 6,, (,, 7,, 8.6. (, 4, 8,, (, 5,,, (, 6,, 4, (, 7,,, (,,, 6, (,, 4, 6, (,, 4, 7, (,, 5, 9, (,, 6,, (,, 7,, (, 4, 8, 5,.74. (, 4,, 9, (, 5,,, 5.6. (, 7, 5,, (, 9,, 9, 8 * The hree rows of each cell shows he ARL, he DRL, and he ercenles ( ρ 5, ρ 5, ρ 5, ρ 75, ρ 95, resecvely. ee A rogram 7 n Aendx B for he calculaon of he values n Table.8. 5

43 Table.8 connued for h, 4,..., 4 and 5, 7,..., h (,, 7, 4, (, 4, 9, 7, 6 (, 5,,, 44 (, 6,, 6, 55 (, 7,,, 68 (, 9,, 4, 86 (,, 6, 5, (, 4, 9, 7, (, 5,,, (, 6,, 6, (, 7, 7,, 7..5 (, 9,, 4, (,, 9, 57, (,, 8, 75, (4,, 5,, (6,, 7, 4, (8, 4,,, (, 59, 4, 84, (8, 99, 7, 47, (7, 48, 55, 7, (5, 95, 7, 49, (, 5,,, (, 6,, 6, (, 7, 7,, (, 9,, 4, (,, 8, 56, (,, 7, 74, (4,, 5,, 8.4,74 (6,, 7, 4, (8, 4,,, (, 59, 4, 8, (8, 98, 6, 47, (7, 48, 55, 79, 5..5 (5, 95, 79, 48, (, 6,, 6, (, 7, 7,, (, 9,, 4, (,, 8, 56, (,, 7, 74, (4,, 5,, (6,, 7, 4, (8, 4,,, (, 59, 4, 8, (8, 98, 6, 47, (7, 47, 54, 78, (5, 94, 78, 45, (, 7,,, (, 9,, 4, (,, 8, 55, (,, 7, 7, (4,, 5,, 5..6 (6, 9, 7, 4, (8, 4,,, (, 59, 4, 8, (8, 98, 5, 469, (7, 47, 5, 75, (5, 9, 74, 48, (, 9,, 4, (,, 8, 55, (, 5, 6, 7, (4,, 49, 98, (6, 9, 69, 8, (8, 4, 99, 98, (, 58, 4, 79, (8, 97,, 464, (6, 46, 5, 7, (5, 9, 699, 97, (,, 7, 5, (, 5, 6, 7, (4,, 48, 96, (6, 9, 68, 6, (8, 4, 98, 95, (, 58, 8, 75, (7, 95, 9, 458, (6, 44, 46, 69, (5, 86, 69, 79, (, 5, 5, 68, (4,, 47, 94, (5, 8, 66,, (8, 4, 95, 9, (, 56, 5, 69, (7, 9, 4, 448, (6, 4, 4, 679, (5, 8, 675, 5, 9 54

44 Table.8 connued for h, 8,..., 8 and,,..., (,,, 5, h (,,, 6, (,, 4, 6, (,, 4, 7, 4 (,, 4, 8, 5 (,, 5, 8, 7 (,, 5, 9, 8 (,,, 6, (,, 4, 7, (,, 5, 8, (,, 5,, (,, 6,, (, 4, 8, 5,..46 (, 4, 9, 8, (, 5,,, (, 6, 5, 8, (, 8, 9, 7, (,, 5, 49, (, 4,, 66, (,, 4, 7, (,, 4, 8, (,, 5, 9, (,, 6,, (,, 7, 4, (, 4, 9, 7, (, 5,,, (, 6, 4, 7, (, 8, 8, 5, (,, 4, 46, (,,, 6, (4, 8, 4, 85, (,, 4, 7, (,, 5, 9, (,, 6,, (,, 7,, (, 4, 8,, (, 5,,, (, 6,, 5, (, 7, 7,, (,,, 4, (,, 9, 58, (4, 7, 4, 8, (5, 4, 57,, (,, 5, 8, (,, 5,, (,, 6,, (, 4, 8, 5,.6.5 (, 4,, 8, (, 5,,, (, 7, 5,, (, 9,, 4, (,, 7, 5, (,, 7, 7, (5,, 5, 4, (6,, 76, 5, (,, 5, 9, (,, 6,, (, 4, 7,, 8..8 (, 4, 9, 7, (, 5,,, (, 6, 4, 7, (, 8, 8, 6, (,, 4, 48, (, 4, 4, 66, (4,, 47, 94, (6, 9, 68, 6, (8, 4, 99, 98, (,, 6,, (,, 7,, 5..6 (, 4, 8, 5, (, 5,, 9, (, 6,, 4, 5.6. (, 7, 7,, (,,, 4, (,,, 59, (4, 8, 4, 8, (5, 6, 6,, (7, 7, 88, 75, (, 6, 4, 8, (,, 6,, (, 4, 7,, 8.4. (, 4, 9, 7, (, 5,,, (, 7, 5, 8, (, 9, 9, 7, (,, 6, 5, (4,, 7, 7, (5,, 5, 4, (7,, 77, 5, (, 5,, 4, (5, 8, 9, 8, 8 55

45 Table.8 connued for h, 8,..., 8 and 7, 9,..., h (4, 9, 45, 9, (5, 7, 64, 7, (7, 8, 9, 8, (, 55,, 6, (7, 9, 7, 4, (5, 8,, 66, (49, 7, 654, 7, (5, 6, 6,, (7, 7, 88, 74, (, 5, 5, 5, (, 86, 7, 44, (4,, 8, 65, (47, 59, 64, 47, (7, 4, 8,, (9, 49, 8, 6, (5, 8, 95, 88, (, 5,, 6, (44, 4, 584, 8, (9, 46, 9, 8, (4, 75, 79, 58, (,, 78, 556, (4,, 55, 8, (, 67,,, (9, 5, 5, 5, (6, 98, 476, 95, (7, 9,, 444, (, 7, 4, 84, (7, 45, 47, 69,

46 Table.8 connued for h,,..., 4 and,,..., h (,, 6,, (,, 6,, (, 4, 7,, (, 4, 7,, 5 (, 4, 8,, 7 (, 4, 8, 4, (,, 7,, (, 4, 8, 5,.96.7 (, 5,, 9, (, 6,, 4, (, 8, 7,, (,,, 44, (, 4,, 6, (4, 9, 45, 88, (6, 8, 65, 9, (9, 4,, 4, (, 67,, 9, (, 9, 84, 567, (, 4, 7,, (, 5, 9,, (, 6,,, (, 7, 5, 8, (, 9, 9, 7, (,, 7, 5, (4,, 8, 74, (5,, 54, 7, (7, 6, 84, 7, (, 55,, 6, (8, 95, 7, 45, (, 4, 8, 4, (, 5,, 8, (, 6,, 4, (, 8,,, (,,, 4, (4, 4,, 6, (5, 9, 45, 88, (6, 9, 68, 4, (9, 44, 5, 9, (4, 74, 77, 5, (, 5, 9,, (, 5,,, (, 7, 4, 6, (, 9, 9, 6, (,, 6, 5, (4,, 6, 7, (6,, 54, 7, (8, 5, 8, 4, (, 57, 6, 7, (, 5, 9, 7, (, 6,,, (, 7,, 9, (,,, 4, (4,, 9, 57, (5, 9, 4, 84, (6, 8, 64, 7, (9, 44,, 5, (, 5,, 8, (, 7,, 4, (, 8, 7,, (,, 4, 45, (4, 5, 4, 65, (5,, 5, 98, (8,, 78, 54, (, 5, 9, 5, (, 6,,, (, 7, 4, 6, (, 9, 9, 6, (4,, 7, 5, (5, 7, 8, 75, (6, 5, 58, 5, 46 57

47 Table.8 connued for h 44, 46,..., 54 and,,..., (, 5, 9,, h (, 5,, 8, 5 (, 6,, 9, 8 (, 6,,, 4 (, 7,,, 4 (, 6,,, (, 8, 5, 9, (,,, 4, (4, 4,, 57, (5,, 44, 86, (, 7,,, (, 8, 7,, (4,,, 44, (5, 5,, 64, (, 7, 4, 5, (, 9, 8, 4, (4,, 5, 48, (, 8, 5, 7, (4,,, 7, (, 8,, 8, (, 7,,, 45 n( n Recall ha he reason why here are so many oen cells s because he values of h s aen o sasfy h. For examle, for n( n ( he reference value h s aen o be smaller han or equal o 44, snce

48 (, (4, (h, (, (4, (4, (, 5 (4, (, 'mall' 'Moderae' 'Large' (h, Fgure.5. Boxlo-le grahs for he n-conrol run lengh dsrbuon of varous uer onesded CUUM sgned-ran chars when n. The whsers exend o he 5 h and he 95 h ercenles. The symbols, and denoe he ARL, DRL * and MRL, resecvely. * For ease of nerreaon, he sandard devaon (as measure of sread s ncluded n he (locaon measures of ercenles. The boxlos are classfed no caegores, namely small ( h 5, moderae ( 5 < h 5 and large ( h > 5. 59

49 Examle.5 An uer one-sded CUUM sgned-ran char for he Mongomery ( son rng daa We conclude hs sub-secon by llusrang he uer one-sded CUUM sgned-ran char usng he Mongomery ( son rng daa. Recall ha he daase conans 5 samles (each of sze 5. For llusraon ae and h 8. From Table. can be seen ha he nconrol average run lengh equals 8. when ( h, (8,. Generally, one chooses he char consans so ha a secfed n-conrol average run lengh, such as 5, or 7, s obaned. Tang hs no consderaon, an n-conrol average run lengh of 8. s consdered small. Recall ha unless he samle sze n s or more, he sgned-ran char s somewha unaracve (from an oeraonal on of vew n C alcaons. The long sascs for he hewhar sgned-ran char ( R for,,..., 5 are gven n he second row of Table.9. The uer one-sded CUUM long sascs ( for,,..., 5 are gven n he las row of Table.9. Table.9. R and values for he son rng daa n Mongomery ( *. amle No: R To llusrae he calculaons, consder samle number. The equaon for he long sasc s max[, R ] max[, 8 ] max[,5] 5 where a sgnalng even occurs for he frs such ha h, ha s, 8. The grahcal dslay of he uer onesded CUUM sgned-ran char s shown n Fgure.6. * The values n Table.9 were generaed usng Mcrosof Excel.

50 Fgure.6. The uer one-sded CUUM sgned-ran char for he Mongomery ( son rng daa. The uer one-sded CUUM sgned-ran char sgnals a samle 7, ndcang a mos lely osve devaon from he nown arge value θ. The acon aen followng an ou-ofconrol sgnal on a CUUM char s dencal o ha wh any conrol char. A search for assgnable causes should be done, correcve acon should be aen (f requred and, followng hs, he CUUM s rese o zero.... Lower one-sded conrol chars The me ha he rocedure sgnals s he frs me such ha he fne-sae Marov chan eners he sae ς where he sae sace s gven by Ω { ς, ς,..., ς r s} wh h ς <... < ς rs, and { h, mn{, R } max. (.4