Extreme Value Charts and Anom Based on Inverse Rayleigh Distribution

Transcription

1 Extreme Value Charts ad Aom Based o Iverse Raylegh Dstrbuto B. Srvasa Rao R.V.R & J.C College of Egeerg Chowdavaram Gutur Adhra Pradesh, Ida boyapatsru@yahoo.com J. Pratapa Reddy St.A's College for Wome Goratla, Gutur-505 Adhra Pradesh, Ida jpstatstcs@yahoo.com K. Rosaah Acharya Nagarjua Uversty Gutur Adhra Pradesh, Ida Rosaah1959@gmal.com Abstract A measurable qualty characterstc s assumed to follow Iverse Raylegh dstrbuto. Varable cotrol charts based o the extreme values of each subgroup are costructed. The techque of aalyss of meas (ANOM) s adopted to work out the decso les of Iverse Raylegh dstrbuto. The preferablty of the proposed ANOM decso les over that of Ott (1967) [11] s llustrated by some examples. Keywords: Extreme Value charts, I Cotrol, Q-Q plot. Acroyms IRD: Iverse Raylegh Dstrbuto f(x) : probablty desty fucto (pdf) F(x): cumulatve dstrbuto fucto (cdf) σ : scale parameter ANOM: aalyss of meas 1. Itroducto The probablty desty fucto (pdf) of a Iverse Raylegh Dstrbuto (IRD) wth scale parameter σ s gve by (- /) x f () x e ; x 0, 0 (1.0.1) x Its cumulatve dstrbuto fucto (cdf) s / F () x e x ; x 0, 0 (1.0.) Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

2 B. Srvasa Rao, J. Pratapa Reddy, K. Rosaah I order to costruct a cotrol chart usg the extreme observatos of a subgroup draw from the producto process wth the qualty varate followg IRD we eed the percetles of extreme order statstcs from IRD. Specfcally, the test statstc o extreme value cotrol chart s the orgal sample vector X(, x1 x,...,) x from the ogog producto. I ths chart all the dvdual sample observatos are plotted to cotrol chart wthout calculatg ay statstc out of them. A correctve acto s take after a sample depedg solely o the extreme values amely x 1 (sample mmum) ad x (sample maxmum) of the sample. Because of ths, the chart s called extreme value cotrol chart. Aalyss of Meas (ANOM) s a techque orgally developed by Ott (1967) [11] for comparg a group of treatmet meas to see f ay oe of them dffers sgfcatly from the overall mea. Ths procedure s carred out by comparg the sample mea values to the overall grad mea, about whch decso les have bee costructed. If a sample mea les outsde these decso les t s declared sgfcatly dfferet from the grad mea. A ANOM chart, coceptually smlar to a cotrol chart, portrays decso les so that statstcal sgfcace as well as practcal sgfcace of samples may be assessed smultaeously. For usg the ANOM techque the cocept of the cotrol chart for meas s vewed a dfferet way groupg of plotted meas to fall wth the cotrol lmts or some outsde the cotrol lmts. For the homogeety of all the meas, t s ecessary that all the meas should fall wth the cotrol lmts. We make a attempt to develop the ANOM procedure of Ott (1967) whe the data varate s suppose to follow IRD. If (1) s take as the cofdece coeffcet we should have the probablty of all the subgroup meas to fall wth the cotrol lmts s (1). Assumg depedet of subgroups the above probablty statemet becomes th power of the probablty of a subgroup mea to fall wth the lmts should be equal to (1)..e., I the samplg dstrbuto of x the cofdece terval for x to le betwee two specfed lmts should 1 be equal to (1). The same prcple s adopted the rest of ths paper through IRD. Because of ths paper ams at explorg ANOM usg cotrol lmts of extreme value statstcs we have cosdered oly the cotrol chart aspects but ot ay recetly developed ANOM tables or techques. However, a detaled lterature about ANOM s avalable Rao (005) ad some related works ths drecto are Erck (1976), Schllg (1979), Ohta (1981), Ramg (198), Maso et al. (1989), Bakr (1994), Berard ad Wludyka (001), Wludyka et al. (001), Motgomery (001), Nelso ad Dudewcz ad Nelso (00), Farum (004), Gurgus ad Tobas (004) ad refereces there. The rest of the paper s orgazed as follows. The basc exposure to extreme value cotrol chart s gve Secto. ANOM appled to IRD usg extreme value cotrol charts of IRD s gve Secto followed by llustratve examples Secto 4. Summary ad coclusos are gve secto Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

3 Extreme Value Charts ad Aom Based o Iverse Raylegh Dstrbuto. Extreme Value Charts The gve sample observatos are assumed to follow IRD model. The cotrol les are determed by the theory of extreme order statstcs based o IRD. The cotrol les are to be determed such a way that a arbtrarly chose x of X(, x1 x,...,) x les wth probablty (1 -α) 1/ wth the lmts. Ths ca be formulated as a probablty equalty the followg way. P() x1 L ad P() x U. The theory of order statstcs say that the cumulatve dstrbuto fucto of the least ad hghest order statstcs a sample of sze from ay cotuous populato are [()] F x ad 1 [1()] F x respectvely. where F() x s the cumulatve dstrbuto fucto of the populato. If (1) s the desred at the would be Takg F() x as the CDF of a stadard IRD model (σ = 1), we ca get solutos of the two equ atos 1 [1()] F x = ad [()] F x = , whch tur ca be used to develop the cotrol lmts of extreme value chart. The solutos of the above two equatos for = (1) 10 s gve Table.1 deoted as Z(1) ad Z () Table.1: Cotrol Lmts of Extreme value charts Z (1) Z () The values of Table.1 dcates the followg probablty statemet: P ( Z, 1,,...,) (1) () Z Z P ( Z, 1,,...,) (1)0.0015() x Z (.0.) (.0.4) Takg x as a ubased estmate of σ, the above equato becomes P ( D x x D x, 1,,...,) Z (1) Z Where D ad () D. Thus D ad D would costtute the cotrol chart costats for the extreme value charts. These are gve Table. for = (1) 10. Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

4 B. Srvasa Rao, J. Pratapa Reddy, K. Rosaah Table.: Costats of Extreme value charts D D Aalyss of Meas (ANOM) - Iverse Raylegh Dstrbuto Suppose x 1,x,...,x k are arthmetc meas of k subgroups of sze each draw from a IRD model. If these subgroup meas are used to develop cotrol charts to assess whether the populato from whch these subgroups are draw s operatg wth admssble qualty varatos. Depedg o the basc populato model, we may use the cotrol chart costats developed by us or the popular Shewart costats gve ay SQC text book. Geerally we say that the process s cotrol f all the sub group meas fall wth the cotrol lmts. Otherwse we say the process lacks cotrol. If α s the level of sgfcace of the above decsos we ca have the followg probablty statemets. P{ LCL x, 1 to k UCL} 1 (.0.6) Usg the oto of depedet subgroups (.0.6) becomes 1 k P { L C L x U C L }(1) (.0.7) Wth equ taled probablty for each subgroup mea, we ca fd two costats say L ad U such that 1(1) k P{ x L } P{ x U } I the case of ormal populato L ad U satsfy U L. For the skewed populatos lke IRD we have to calculate L, U separately from the samplg dstrbuto of x. Accordgly these deped o the subgroup sze ad the umber of subgroups k. We make us of the equatos (.07) ad (.08) for specfed ad k to get L ad U for α = 0.01 ad α = These are gve Tables. ad.. A cotrol chart for averages gvg I Cotrol cocluso dcates that all the subgroup meas though vary amog themselves are homogeeous some sese. Ths s exactly 76 Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

5 Extreme Value Charts ad Aom Based o Iverse Raylegh Dstrbuto the ull hypothess a aalyss of varace techque. Hece the costats of Tables. ad. ca be used as a alteratve to aalyss of varace techque. For a ormal populato oe ca use the tables of Ott (1967) [11]. For a IRD our tables ca be used. We therefore preset below some examples for whch the goodess of ft of IRD model assessed wth Q Q plot techque (stregth of learty betwee observed ad theoretcal quatles of a model) ad tested the homogeety of meas volved each case. 4. Illustratve Examples Example 1: Wadsworth ( 1986): Cosder the followg data of 5 observatos o A maufactures of metal products that suspected varatos ro cotet of raw materal suppled by fve supplers. Fve gots were radomly selected from each of the fve supplers. The followg table cotas the data for the ro determatos o each gots percet by weght. Supplers Example : Three brads of batteres are uder study. It s suspected that the lfe ( weeks) of the three brads s dfferet. Fve batteres of each brad are tested wth the followg results. Test whether the lves of these brads of batteres are dfferet at 5 % level of sgfcace. Weeks of lfe Brad 1 Brad Brad Example : Four catalysts that may effect the cocetrato of oe compoet a three compoet lqud mxture are beg vestgated. The followg cocetratos are obtaed. Test whether the four catalysts have the same affect o the cocetrato at 5 % level of sgfcace. Catalyst Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

6 B. Srvasa Rao, J. Pratapa Reddy, K. Rosaah The goodess of ft of data these three examples as revealed by Q Q plot (correlato coeffcet) are summarzed the followg table, whch shows that IRD s a better model, exhbtg sgfcace lear relato betwee sample ad populato quatles. IRD Normal Example Example Example Treatg these observatos the data as a sgle sample, we have calculated the decso lmts for the Normal populato as well as verse Raylegh populato ad have gve them the Table.4 respectvely. Table.4 No. of subgroups fall (LDL, UDL) Wth the decso les Coverage probablty Outsde the decso les Coverage probablty Example 1 = 5, k = 5, α =0.05 [.79,.517] [5.699, 9.57] Example = 5, k =, α =0.05 [ 87.8, 95.5 ] [ , 91.64] Example = 4, k = 4, α =0.05 [6.14, 8.84] [ , ] I each cell the frst row values represets the Normal dstrbuto ad secod row values represets the Iverse Raylegh dstrbuto. 5. Summary ad Coclusos ANOM tables of Ott (1967) [11] yeld that the umber of homogeeous meas for each data set are,, respectvely ad those away from heterogeety are,1, respectvely. O the other had whe the ANOM tables of our model (IRD) are used for data sets we get the umber of homogeeous meas to be 5,,4 respectvely wthout exhbtg devato of ay mea from homogeety. Thus usage of ormal model resulted homogeety for some meas ad devato from some other meas, dcatg a possble rejecto of these meas. Ths decso s vald f Normal dstrbuto s a good ft to the data. As a comparso, we have already establshed by Q-Q plot that IRD s a better model tha Normal as supported by the Q-Q plot correlato coeffcet of each data set wth Normal as well as IRD separately. Therefore, all the meas to be homogeeous wth the help of IRD (Table.5) s a better decso tha some meas to be away from homogeety usg Normal, ANOM procedure. 764 Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

7 Extreme Value Charts ad Aom Based o Iverse Raylegh Dstrbuto Refereces 1. Bakr. S.T, Aalyss of meas usg raks for radomzed complete block desgs, Commucatos Statstcs- Smulato ad Computato (1994), Berard. A.J. ad Wludyka. P.S, Robust -sample aalyss of meas type radomzato tests for varaces, Joural of Statstcal Computato ad Smulato 69 (001), Dedewcz. E.J. ad Nelso. P.R, Heteroscedastc aalyss of meas (haom), Amerca Joural of Mathematcal ad Maagemet Sceces (00), Erck. N.L, A aalyss of meas a three-way factoral, Joural of Qualty Techology 8 (1976), Farum. N.R, Aalyss of meas tables usg mathematcal processors, Qualty Egeerg 16 (004), Gurgus. G.H ad Tobas. R.D, O the computato of the dstrbuto for the aalyss of meas, Commucato Statstcs- Smulato ad Computato 16 (004), Gust. R.F. Maso. R.L. ad Hess. J.L, Statstcal desg ad aalyss of expermets, Joh Wley ad Sos, New York, Motgomery. D.C, Desg ad aalyss of expermets, ffth ed., Joh Wley ad Sos, Nelso, P.R. Wludyka. P.S. ad Slva. P.R, Power curves for aalyss of meas for varaces, Joural of Qualty Techology (001), Nelso. P.R. ad Dudewcz. E.J, Exact aalyss of meas wth uequal varaces, Techometrcs 44 (00), Ohta. H, A procedure for poolg data by aalyss of meas, Joural of Qualty Techology 1 (1981), Ott. E.R, Aalyss of meas- a graphcal procedure, Idustral Qualty Cotrol 4 (1967). 1. Ramg. P.F, Applcato of aalyss of meas, Joural of Qualty Techology 15 (198), Rao. C.V ad Prakumar. M, Aom-type graphcal methods for testg the equalty of several correlato coeffcets, Gujarat Statstcal Revew 9 (00), Rao. C.V, Aalyss of meas- a revew, Joural of Qualty Techology 7 (005). 16. Schllg. E.G, A smplfed graphcal grat lot acceptace samplg procedure, Joural of Qualty Techology 11 (1979), Stephes. K.S. Wadsworth. H.M. ad Godfrey. A.F, Moder methods of qualty cotrol ad mprovemet, Joh Wley ad Sos. Ic. New York, d ed Pak.j.stat.oper.res. Vol.VIII No.4 01 pp

8 B. Srvasa Rao, J. Pratapa Reddy, K. Rosaah Table.: Iverse Raylegh Dstrbuto Costats for Aalyss of Meas (1-α = 0.99) k = Table.: Iverse Raylegh Dstrbuto Costats for Aalyss of Meas (1-α = 0.95) Pak.j.stat.oper.res. Vol.VIII No.4 01 pp