Jurnal Teknologi. A Test for Normality in the Presence of Outliers. Full paper. t t u. Pooi Ah Hin a*, Soo Huei Ching a

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1 Jrnal Tenolog Fll paper A Te for oraly n he Preence of Oler Poo Ah Hn a Soo He Chng a a Snway Unvery Bne School o 5 Jalan Unver Bandar Snway 650 Pealng Jaya Selangor Malaya Correpondng ahor: ahhnp@nway.ed.y Arcle hory Receved : Janary 0 Receved n reved for : 7 May 0 Acceped :5 Jne 0 Graphcal abrac V Abrac The JarqeBera e a e baed on he coeffcen of ewne (S) and ro (K) for eng wheher he gven rando aple fro a noral poplaon. When he rando aple of ze n conan oler we e he reanng n obervaon o cope wo ac S and K whch c he ac S and K. The ac S and K are nex ranfored o z and z whch are ncorrelaed and havng andard noral drbon when he orgnal poplaon noral. We how ha he accepance regon gven by a crcle n he (z z ) plane able for eng he noraly apon. Keyword: Qadracnoral drbon; nonlnear correlaon; ehod baed on oen; crclar accepance regon; power of e Abra Ujan JarqeBera adalah a jan yang berdaaran peal epencongan (S) dan ro (K) n engj aada apel rawa yang dber beraal dar a popla noral. Bla apel rawa dengan az n engandng daa erpencl n cerapan yang ernggal dgnaan n enghng da a S dan K yang enr a S dan K. Sa S dan K edan djelaan epada z dan z yang da berorela dan berabran noral pawa bla popla aal adalah noral. Ddapa bahawa awaan peneraan yang berdaaran a blaan dala aah (z z ) adalah ea n engj enoralan daa. Kaa nc: Tabran adranoral; orela a lnear; aedah berdaaran oen; awaan peneraan yang berben bla; aa jan 0 Penerb UTM Pre. All rgh reerved..0 ITRODUCTIO The valdy of any acal procedre depend on noraly apon of obervaon. Several ehod have been nrodced n he lerare for aeng he apon of noraly. The ore poplar one aong he are he Jarge Bera e n [] ShaproWl e n [] and Anderon Darlng e n []. The JargeBera (JB) e for noraly baed on he aple coeffcen of ewne and ro he Shapro Wl e rele on he correlaon n he QQ plo and he AnderonDarlng e ae e of he dfference beween eprcal and heorecal drbon. The JB e ha been odfed by a nber of ahor [ 6]. Gel and Gawrh [] replaced he denonaor of ewne and ro n he JB e ac by he average abole devaon fro he edan (MAAD) o oban he Rob JargeBera (RJB) e ac. In regreon analy he eae of he coeffcen of ewne and ro baed on he Ordnary Lea Sqare (OLS) redal have been recaled by Ion [5] and he relng JB ype e called he Recalled Moen (RM) e. By replacng he eae of pread wh MAAD he RM e can be odfed o yeld ye anoher e called he Rob Recaled Moen (RRM) e n [6]. Preenly we conder he proble of aeng noraly gven a rando aple y y... yn of ze n and conanng oler. We ae ha he oler are no de o carele ae. Inead we ae ha he ole are de o defcence n he earng ye n earng n of whch he arbe o be eared have very large or all vale. For exaple a prng balance ay rech dproporonaely when he objec o be weghed very heavy. We h ae ha he exreely large (or all) vale whch are clafed a oler n he gven aple are orgnally very large (or all) vale. However de o defcence n he earng ye hey are dored o oe exreely large (or all) vale whch are clearly no concordan wh he re of he daa. Wh h apon we defne he h adjed aple oen by n y M = n 6: (0) eiss 80 7 ISS

2 96 Poo Ah Hn & Soo He Chng / Jrnal Tenolog (Scence & Engneerng) 6: (0) where he nber of oler a he lower end he nber of oler a he pper end = + and M he aple edan baed on he orgnal aple of ze n. Hence he adjed coeffcen of ewne and ro can be defned repecvely a S and K. The ac S and K are nex expreed repecvely a nonlnear fncon of he ncorrelaed rando varable z and z whch have andard noral drbon when he orgnal poplaon noral. We how ha he accepance regon gven be a crcle wh cenre zero n he (z z) plane able for eng he noraly apon. The layo of he paper a follow. In Secon we ae he JargeBera e. In Secon we nrodce a ype of nonnoral drbon called he qadracnoral drbon nrodce by [7]. In Secon we decrbe he ehod n [8] whch ranfor a e of nonlnearly correlaed nonnoral rando varable o a e of ncorrelaed andard noral rando varable. In Secon 5 he ehod n Secon wll be ed for ranforng he ac S and K whch c he coeffcen of ewne (S) and ro (K) o he ncorrelaed andard noral varable z and z. The power of he e baed on a crcle n he (z z) plane derved fro (S K) when here are oler nex copared wh he power of he e baed on a crcle n he (z z) plane derved fro (S K) when here are no oler and alo wh he power of he JB e. In Secon 6 we gve oe concldng rear..0 MATERIALS AD METHODS. JarqeBera Te Gven he rando aple y ( y y... y ) T n he aple coeffcen of ewne and ro are gven repecvely by S and K n n where y y ac gven by S JB σ and K σ S K n y n y. The JB e where σs 6 n and σk n are repecvely he aypoc varance of S and K. When he y are norally drbon he JB ac ha approxaely a ch qare drbon wh degree of freedo. The rejecon regon baed on he above ch qare approxaon for he hypohe of noraly gven by y: JB χ where χ he 00( α)% pon of he ch qare drbon wh degree of freedo.. Qadracoral Drbon Le μ ( ) T be conan and conder he followng nonlnear ranforaon μ z z y z 0 = μ z z z 0 where z ha he andard noral drbon and y a oneoone fncon of z. The rando varable y hen ad o have he qadracnoral drbon wh paraeer and a ndcaed n [7]. We ay wre y ~ Q( μ ).. Tranforng Correlaed onoral Varable o Uncorrelaed Sandard oral Varable Le (... ) T be a e of correlaed nonnoral rando varable. Sppoe we have a e of oberved vale of. T Le he nh oberved vale be denoed by ( n n... n ). The followng a procedre odfed fro he ehod gven n Poo (006) for fng a nonnoral drbon o he oberved vale of : () Fnd he average vale of :. n n () Eae he oen E E( ) E( ) ng ( )( ) ˆ Mj n jn n 0 0 where. j j () Cope he varancecovarance arx ˆ ( )() M j M and fnd he arx V fored by he egenvecor of M. n n n T n () Fnd V. n n (5) Cope he oen ( ~ M )( ) j ~ n ~ jn 0 0. ˆ (6) Fnd n Var n ( )() Var ˆ M. j n where ( )( ) (7) Cope he oen ˆ Mj n jn 0 0. (8) Fro he oen M ˆ ( )( 0) fnd h oen of a rando varable n () ch ha he wh he j

3 97 Poo Ah Hn & Soo He Chng / Jrnal Tenolog (Scence & Engneerng) 6: (0) () Q(0 ) drbon eqal o. M ˆ ( )( 0) (9) Fnd h and h jl j l of whch hjl hlj E of ch ha he heorecal oen j h h h jl j l jj j l j coped by rercng he varance of o one and () ng he approxae drbon Q(0 ) for ˆ j ( )( ) approxaely eqal o M j 0 0. (0) We ay decrbe he drbon of va he eqaon V and a gven n Sep where Var (9) a qadrac fncon of = z z () ( ) ( ) z 0 () ( ) ( ) ( ) z z. z 0 Th he eqaon n Sep (0) provde a ehod for ranforng o ( z z... z ) whch a e of ncorrelaed andard noral rando varable. Table 5. Paraeer of he Drbon of (S K) Paraeer 0L0U 0LU 0LU LU.5E E V V 9.8E V 9.8E V h h Var Var h h 0.0.7E E8 h h h.7e h () () () () () () RESULTS AD DISCUSSIO Conder he cae when he aple ze n = 0 and here are no oler. A oal of = 0000 vale of y are generaed fro a noral drbon. For each generaed vale of y we e he forla n Secon o cope (S K). Le he vale of (S K) baed on he nh generaed of y be denoed by ( ). By applyng he ehod n Secon we can n n ranfor ( ) o ( z z ) whch a e of wo ncorrelaed andard noral varable. We ay denoe he cae wh oler a he lower end and oler a he pper end by L U. When ( ) = (0 ) (0 ) or ( ) we can lewe generae = 0000 vale of y fro a noral drbon cope ( n n ) ( S K) and ranfor ( ) o ( z z ). The vale of he paraeer of he drbon of (S K) are hown n Table 5.. We ay e he rejecon regon ( S K) : z z χ for eng he hypohe of noraly a he 0.05 level. The power of he JB e wh rejecon regon {(S K): JB >.797} and hoe of he e baed on a crcle n he (z z) plane derved fro (S K) are hown n Table 5.. The coln labeled by and gve repecvely he coeffcen of ewne and ro of he varable y.

4 98 Poo Ah Hn & Soo He Chng / Jrnal Tenolog (Scence & Engneerng) 6: (0) Table 5. Power of JB e and e baed on a crcle n he ( z z ) plane (n = 0 α =0.05 = 0000) o. JB 0L0U 0LU 0LU LU

5 99 Poo Ah Hn & Soo He Chng / Jrnal Tenolog (Scence & Engneerng) 6: (0) When (0 ) n whch cae he y are norally drbed Table 5. how ha he power of he JB e and he e baed on a crcle n he (z z) plane are all no oo far fro he arge vale The coln labeled by JB and 0L0U n Table 5. how ha n row 6 he power n he 0L0U coln are larger han hoe n he JB coln by ore han %. Th n he cae when here are no oler he e baed on a crcle n he (z z) plane a rong copeor o he JB e. The coln labeled by 0L0U and 0LU n Table 5. how ha when here an oler a he pper end he e baed on a crcle n he (z z) plane ffer a lgh lo n he power of he e. The coln labeled by 0LU and 0LU how ha when he nber of oler a he pper end ncreae by one he power of he e end o decreae frher. The coln labeled by 0L0U and LU alo how ha here a lgh lo n he power of he e when here are oler on boh end..0 COCLUSIO The e baed on a crcle wh cenre zero n he (z z) plane end o have good power. Th no rprng becae he crcle wh cenre zero he alle regon n he (z z) plane wh he gven probably. When he nber of oler all here a lgh lo n he power of he e. When he nber of oler large lely ha he lo n power wold be large. Reference [] Jarge C. M. and Bera A. K Effcen Te for oraly Hoocedacy and Seral Dependence of Regreon Redal. Econ. Le. 6: [] Shapro S. S. and Wl M. B An Analy of Varance Te for oraly (Coplee Saple). Boera. 5(): [] Anderon T. W. and Darlng D. A. 95. Aypoc Theory of Ceran GoodneOfF Crera Baed on Sochac Procee. Annal of Maheacal Sac. : 9. [] Gel Y. R. and Gawrh J. L A Rob Modfcaon of he JargeBera Te of oraly. Econ. Le. 99: 0. [5] Ion A. H. M. R. 00. Regreon Redal Moen and Ther Ue n Te for oraly. Con. Sa. Theor. Mehod. : 0 0. [6] Rana M.S. Md H. and Ion A. H. M. R A Rob Recaled Moen Te for oraly n Regreon. Jornal of Mah and Sa. 5(): 5 6. [7] Poo A. H. 00. Effec of onnoraly on Confdence Inerval n Lnear Model. Techncal Repor o.6/00. Ine of Maheacal Scence Unvery of Malaya. [8] Poo A. H onnorally Drbed Varae wh a onlnear Dependence Srcre. Techncal Repor o./006. Ine of Maheacal Scence Unvery of Malaya. [9] Ba L. E. Pere T. Sole G. and We A Maxzaon Technqe Occrrng n he Sacal Analy of Probablc Fncon of Marov Chan. The Annal of Maheacal Sac. (): 6 7.