A CONTROL CHART FOR HEAVY TAILED DISTRIBUTIONS. K. Thaga. Department of Statistics University of Botswana, Botswana

Transcription

1 A CONTROL CHART FOR HEAVY TAILED DISTRIBUTIONS K. Thaga Departmet of Statstcs Uversty of Botswaa, Botswaa ABSTRACT Stadard cotrol charts wth cotrol lmts determed by the mea ad stadard error of the mea are costructed based o the assumpto that the dstrbuto of the qualty characterstc beg motored follows a ormal dstrbuto. However, ths assumpto s ot always vald. It s proposed to use a chart based o computg the cotrol lmts usg the process mea ad the stadard error of the least absolute devato for the case where the process qualty characterstcs follow a heavy taled t dstrbuto. Such a cotrol chart s more effectve tha the ormal dstrbuto based chart sce t has a low out-of-cotrol average ru legth for both small ad large values of process shft. OPSOMMING Kotrolekaart wat gebruk maak va kotrolelmete gebaseer op de stadaardafwykg va de gergste absolute lmet word otwerp vr t-verdelg met betekesvolle stert. Smulasetoetse vr vergelykg va de voorgestelde kotrolekaart met ormaalverdeelde kotrolekaarte too dat korter gemddelde looplegtes voor dagose va beheerverles utgewys word, berek word. South Afrca Joural of Idustral Egeerg November 9 Vol ():

2 1. INTRODUCTION Statstcal process cotrol charts such as the Shewhart cotrol chart (Shewhart [8]), the cumulatve sum cotrol chart (Page [5]), ad the expoetally weghted movg average cotrol chart (Roberts [6]), are used to motor product qualty ad detect specal evets that may be dcators of out-of-cotrol stuatos. Such charts are desged o the assumpto that a process beg motored wll produce measuremets that ca be modeled wth a depedet ad detcally dstrbuted ormal dstrbuto, whe oly the heret sources of varablty are preset the system. However, certa applcatos the process may produce measuremets that ca be represeted wth heavy taled dstrbutos. I ths case, the stadard cotrol charts based o ormalty assumptos wll ot rapdly detect out-of-cotrol stuatos, sce the cotrol lmts wll be stretched partcularly whe the ature of the products s such that oe caot take large samples to be able to use the cetral lmt theorem. Several charts based o outler resstat statstcs have bee proposed for use whe there are outlers the process measuremets. These clude, amog others, the charts whose cotrol lmts are calculated usg the meda mdrage ad meda rage by Ferrell [1]. Lageberg ad Iglewcz [4] proposed charts whose cotrol lmts are determed by the trmmed mea of the subgroup meas ad the trmmed mea of the rages. Whte ad Schroeder [11] proposed a chart costructed by plottg subgroup box plots. Such a chart uses the subgroup meda ad subgroup terquartle rage. Rocke [7] proposed a seres of robust cotrol charts that use combatos of subgroup trmmed ad utrmmed mea, meda rage ad terquartle rage. Most of the charts dscussed above use resstat statstcs to determe the cotrol lmts, ad the motor the subgroup meas for the occurrece of out-of-cotrol sgals. The meda charts are less sestve to process shfts sce the meda s ot affected by outlers or extreme value. The chart that plots the mea ad rage wth cotrol lmts determed from the subgroup meas ad the terquartle rages s more effectve detectg mea shfts. For a heavy taled dstrbuto, the extreme observatos are ot ecessarly outlers or sgs of the presece of assgable causes of varato. Thaga [9] proposed a chart based o the least absolute devato that s effectve motorg the process whose qualty measuremet follows a heavy taled dstrbuto. The chart s effectve for motorg processes wth qualty characterstcs that are autocorrelated. It s proposed ad show ths artcle that, for depedet processes, whe process varables follow a heavy taled dstrbuto, a chart where cotrol lmts are determed usg the stadard errors of the least absolute devato estmators performs better tha the chart where cotrol lmts are calculated usg the stadard errors of the ordary least squares estmators. It s also proposed that the rato of mea devato to stadard devato should be used to determe the approprateess of ths chart relato to the stadard ormal dstrbuto based cotrol chart.. LEAST ABSOLUTE DEVIATIONS ESTIMATORS Cosder a regresso model of the form Y h, ) X The ( (1), s are assumed to be depedet for = 1,,..., ad have a symmetrc dstrbuto. A least absolute devato (LAD) estmator of s a soluto to the problem 18

3 m Y h(, ). () X 1 Here the dscrepacy betwee the respose varable Y ad ts approxmato h ( X, ) provded by the model s measured by the L 1 dstace stead of the usual L dstace, whe studyg the least squares estmate. A dffculty wth the least absolute devato method arses from the odfferetablty of the objectve fucto equato (). Ths fucto, however, s cotuous ad cotuously dfferetable at every pot except at zero, where left ad rght dervatves exst. Because of such propertes, a approach smlar to that of Thavaeswara ad Heyde [1] ca be followed, provded that the dervatve at every pot s replaced by the rght dervatve. Ths dervatve s gve by x x I x I x (3).e. the least absolute devato estmatg fucto s gve by glad ~ h( X, ˆ ) ( Y, ) ( I ( (, ˆ I Y ) ) ( (, ˆ h X Y h X ) ) 1 (4) Let f be the codtoal desty fucto of Y/X such that f()>. Uder sutable regularty codtos (Goureroux ad Mofort []), t ca be show that the formato assocated wth the estmatg fucto s 1 I, 4 f () (5) Where I 1 h( X, ˆ ) h( X, ˆ E I a smple lear regresso model, h ( x, ) = x ad y x (6) The Therefore Var ˆ ( LSE ) ad I x. 1 x 1 ~ 1 Var( ), where ˆ s the ordary least square (OLS) 4 f () x 1 estmator ad ~ s the least absolute devato estmator. The least absolute devato estmatg fucto s more effcet tha the ordary least squares estmatg fucto f 19

4 the dstrbuto of the error term s such that 4 f () 1/, where s the varace of the error term. For the case wth ormally dstrbuted errors, the ordary least square estmatg fucto s more effcet tha the least absolute devato estmatg fucto. Whe the errors have a Cauchy dstrbuto, t ca be show that 4 f () (Goureroux ad Mofort []), ad the least absolute devato estmatg fucto s more effcet tha the least squares estmatg fucto. 3. THE NEW CONTROL CHART Whe the observatos are depedet detcally dstrbuted ormal radom varables, the observato at tme t ca be represeted as yt t. The Shewhart cotrol chart for the process mea s developed wth the followg cotrol lmts: LCL θˆ 3 CL θˆ UCL θˆ 3 σ σ, (7) where ˆ s the process mea ad estmatorˆ. s the stadard error of the ordary least square As metoed earler, whe the process measuremets follow a heavy taled dstrbuto, the ordary least square estmator has a greater stadard error tha the least absolute devato estmator. Therefore a cotrol chart based o the stadard error of the ordary least square estmator has wde cotrol lmts. A chart based o the least absolute devato estmator s proposed as follows: The cotrol lmts for the chart are gve as: σ LCL θˆ 3 f () CL θˆ σ UCL θˆ 3 f (), (8) where 1 f () s the stadard error of the least absolute devato estmator. The chart may be costructed by plottg the subgroup meas agast tme or sample umber wth cotrol lmts gve equato (8). If the process measuremets have a t dstrbuto wth v degrees of freedom, the 4 f () ( (( v 1) / )) 4 v ( ( v / )) (9) 11

5 It The varace of the error term s fte for v 3 ad s gve by 1/ ( v ) / v. ca be show that for v = 1, the varace s fte ad 4 f () = 4 /. v = the varace s fte ad 4 () Smlarly, for f =.5 (Thavaeswara ad Heyde [1]). Ths shows that for heavy taled t dstrbutos wth fte varace, the least absolute devato approach s superor. It ca also be show that for v = 3, 4 f (). 54 ad for v = 4, 6. I these cases, the least absolute devato estmatg fucto provdes more formato tha the ordary least squares estmatg fucto. For a t dstrbuto wth v 5 wth th tals the least squares estmatg fucto provdes more formato. Table 1 shows the cotrol lmts for data smulated for a process that follows a t dstrbuto wth varous degrees of freedom. 15, subgroups of three observatos each have bee smulated. The statstc x s the sample ordary least square estmator, ad x ~ s the sample least absolute devato estmator. It ca be see that the least absolute devato estmator has a small stadard error whe the process produces measuremets that follow a heavy taled t dstrbuto, whle the ordary least square estmator has a small stadard error for a t dstrbuto wth fve or more degrees of freedom. Therefore, for heavy taled data, the cotrol lmts for charts based o the ormalty assumpto are wder tha those computed usg the least absolute devato estmator. Wder cotrol lmts result the cotrol chart ot beg able to detect process shfts rapdly partcularly small shfts. Therefore the chart based o ordary least square estmators s ot recommeded for heavy taled dstrbutos. df. x x ~ s.e( x ) s.e( x ~ ) LCL UCL LCL UCL OLS LAD Table 1: Cotrol lmts for the charts based o OLS ad LAD estmators for a process followg t dstrbutos. To decde whch procedure to use, t s recommeded that oe should frst calculate the rato of the mea absolute devato to the stadard devato. Whe the process qualty characterstc follows a ormal dstrbuto, oe expects ths rato to be.77. However, sce the least absolute devato estmator performs better tha the ordary least square estmator for a t dstrbuto wth four or less degrees of freedom, t s recommeded that the chart based o the least absolute devato estmator be used whe the rato s.77 or less. Whe the qualty characterstc follows a t dstrbuto wth four degrees of freedom, the rato of the mea devato to the stadard devato s.77. For a t dstrbuto wth three degrees of freedom, the rato s.637 (Johso ad Kotz [3]). The average ru legths, whch are the average umber of pots that must be plotted by the chart before a pot falls beyod the cotrol lmts, are commoly used statstcal process cotrol measures for comparg the performace of cotrol charts. Whe a pot falls outsde the cotrol lmt(s), the chart ssues a out-of-cotrol sgal, dcatg the possble presece of a assgable cause of varato the process. It s therefore desrable for a chart to have a small average ru legth whe the process has shfted, ad a large average ru legth whe the process s cotrol. 111

6

7

8

9 [3] Johso, N. L. ad Kotz, S Cotuous uvarate dstrbutos, Vols. 1 &. Houghto Mffl Compay, Bosto. [4] Lageberg, P. ad Iglewcz, B Trmmed mea X ad R charts, Joural of Qualty Techology, 18, [5] Page, E.S Cumulatve sum charts. Techometrcs, 3, 1-9. [6] Roberts, S.W Cotrol charts test based o geometrc movg averages. Techometrcs, 1, [7] Rocke, D.M Robust cotrol charts. Techometrcs, 31, [8] Shewhart, W.A Ecoomc cotrol of qualty of maufactured product. Va Nortrad Ic., New York. [9] Thaga, K. 8. Cotrol chart for autocorrelated processes wth heavy taled dstrbutos. Ecoomc Qualty Cotrol. 3, 1-1. [1] Thavaeswara, A. ad Heyde, C.C Predcto va estmatg fuctos. Joural of Statstcal Plag ad Iferece, 77, [11] Whte, E.M. ad Schroeder, R A smultaeous cotrol chart. Joural of Qualty Techology, 19,