Testing for multivariate normality of disturbances in the multivariate linear regression model Yan Su a, Shao-Yue Kang b

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1 Itertol Coferece o Itellget Systes Reserch d Mechtrocs Egeerg (ISRME 05) Testg for ultvrte orlty of dsturbces the ultvrte ler regresso odel Y Su, Sho-Yue Kg b School of Mthetcs d hyscs, North Ch Electrc ower Uversty, Bodg, 07003, Ch el: suyhd@63co, b el:kgsy5@63co Keywords: Ler odel; Dsturbces; Resduls; Multvrte orl dstrbuto; Goodess-of-ft Abstrct We suggest chrcterstc test for testg the ultvrte orl dstrbuto of the dsturbces the ultvrte ler regresso odel(mlrm) The test s bsed o the goodess-of-ft test for uforty o the surfce of ut sphere The syptotc ull dstrbuto of the trsfored resduls fro the MLRM s obted A lgorth s gve to pproxte the crtcl vlues of the test by Mote Crlo sulto The test possesses syetry d c be esly coputed for rbtrry deso of the dsturbce vectors Itroducto A ultvrte ler odel descrbes the reltoshp betwee respose vector y d vector x of covrbles Let y,, y be depedet observto vectors R, followg the odel y = xβ + ε, =,,, () E( ε ) = 0,Cov( ε ) = Σ, () p where the pre deotes trspose, the desg vectors x R re ssued to be ordo, β s ukow p trx of preters clled regresso coeffcets, ε, re the -vectors of dsturbces (or errors), d Σ s ukow postve defte trx Clsscl theory o the ultvrte ler odel ssue the dsturbces ε () re orlly dstrbuted To vod wrog coclusos regresso lyss, the dstrbutol ssupto o the dsturbces should be checked Let F be the ukow dstrbuto of the dsturbces ε d let F0 be the N(0, Σ) dstrbuto We wt to test the hypothess F= F 0 (3) Bsed o tegrl of the squred odulus of the dfferece betwee the eprcl chrcterstc fucto of the resduls d the chrcterstc fucto uder the ull hypothess, Gero, Grcí d Meís(005) proposed goodess-of-ft test for y fxed dstrbuto of dsturbces ultvrte ler odels[] Let Ω deote the surfce of ut sphere cetered t the org R d let U ( Ω ) deote the ufor dstrbuto o Ω Bsed o the oet of ert d the ceter of ss of the sples o Ω, Su d Yg (009) proposed the goodess-of-ft tests for U ( Ω ), the power sulto hs show tht the test hs good power[] Su d Yg(0) exteded the test [3], we preseted test for uforty dstrbuto o the surfce of ut sphere bsed o geerlzed verse, the test possesses syetry d hs ce propertes[3] Hece, the test c crese the test power Let ˆ ε be the resduls of the ultvrte ler regresso odel The syptotc ull dstrbuto of the trsfored resduls s U ( Ω ) Therefore, the goodess-of-ft test for the ultvrte orl dstrbuto of the dsturbces ε () c be trslted to the 05 The uthors - ublshed by Atlts ress 40

2 goodess-of-ft test for U ( Ω ) Bsed o the goodess-of-ft test for U ( Ω ), ths pper troduces ew test for the ultvrte orl dstrbuto of the errors ε () The trsforto bsed o Cholesky decoposto leds to the trsfored resduls whose ot dstrbuto syptotclly does ot deped o the ukow preter Σ of the N(0, Σ) dstrbuto Thus, the crtcl vlues of the test sttstc c be estted by Mote Crlo ethods wth Σ = I,where I deotes the detty trx The pper s orgzed s follows I Secto, we troduce the ultvrte ler regresso odel d soe les I Secto 3, the chrcterzto-bsed test for ultvrte orl dsturbces s proposed The syptotc ull dstrbuto of the trsfored resduls s obted I Secto 4, the lgorth to estte the crtcl vlues s gve Soe dscussos d further pplctos re gve Secto 5 The ultvrte ler odel d soe les Let Y = ( y,, y ), X = ( x,, x ), ε = ( ε,, ε ) (4) The the ultvrte ler odel ()-() tkes the for Y = Xβ + ε, (5) E[ vec( e )] = 0, Cov[ vec( e )] = I Σ, (6) wherey d ε re rdo trces, X s kow ptrx, d β s ukow p trx Here, the sg deotes the kroecker product of trces The stteet tht the rdo trx ε N (0, I Σ ) s equvlet to the stteet tht the rdo vector vec( e ) N (0, I Σ ) The ultvrte ler odel (5) -(6) geerlzes the ultple ler odel ( = ) by llowg vector of observtos, gve by the rows of trx Y, to correspod to the rows of the desg trx X Le [4] Let the odely = Xβ + ε be defed (5) Let rk ( X) = pd let ε N (0, I Σ ), X = X( XX) X, (7) Let ˆβ be the xu lkelhood estte of β, e, ˆ=( β X X) XY Let ˆ ˆ ε = ( ˆ ε ˆ, ε) = Y Xβ, ˆ Σ = ˆ εεˆ, l XX= D, (8) p - where D s postve defte trx The () βˆ d ˆΣ re depedet Moreover, ˆ β β, Σ Σ ˆ,, (9) where deotes covergece probblty s (b) The resdul trx ˆ ε = ( I X )ε, ˆ ε ~ N(0,( I X) Σ ) (0) Moreover, the th row of ˆε, deoted s ˆ ε, hs -vrte orl dstrbuto, e, ˆ ε ~ N(0, ( h ) Σ ), =,,,, () where h dctes the th eleet of X (7) Le [5] (The Cholesky decoposto) If A s postve defte trx the there exsts uque lower-trgulr trx L wth postve dgol eleets such tht A = LL Le3 [5] Let ε N (0, I Σ) d let Σ = εε/ ( p) Let the Cholesky decoposto of Σ be Σ= [ L( Σ )][ L( Σ )] d let w = [ L( Σ )] ε, =,,, W = ( w,, w ), () 4

3 whereε s defed (4) The the dstrbuto of W does ot deped o Σ Defto A trxc such thtcc C = C s clled geerlzed verse of C The Moore- erose C + s geerlzed verse of trxc tht stsfes the followg requreets: CC C = C,C CC = C,(C C) = C C,(CC ) = CC Defto [6] LetU U( Ω ) A rdo vectorς s sd to hve sphercl dstrbuto f ς hs stochstc represetto ς d = η U for soe rdo vrbleη 0 of U Here = d sgfes tht the two sdes hve the se dstrbuto Le4 [6] If A rdo vectorς hs sphercl dstrbuto the where deotes the Euclde or ς ς U ( Ω ), Le5 [6] LetU = ( U,, U ) U( Ω ) The, whch s depedet EU ( ) = 0,Cov( U ) = I Rerk The covrce trx Cov( U ) = ( / ) I s the chrcterzto of U ( Ω )The Cov( U ) of U correspods to the oet of ert of U (e, the secod oet of the coordte vrble of U ) Le6 [] Letτ = ( τ,, τ ) be rdo vector Let τ = d let The the expectto oet of ert of deped o where H f d oly f H = ττ τ bout rbtrry drecto E( τ ) =, =,,, E( ) = 0,,, =,,, Cov τ ( ) ( ) H ( h,, h ) exst = does ot Rerk Le6 dctes tht the expectto oet of ert of U bout rbtrry drecto s the se f U s uforly dstrbuted o Ω Le7 [3] Let U = ( U,, U ) U( Ω ) d let G = ( U,, U ), = (/,,/ ) LetU = ( U,, U ), =,, be d U ( Ω ) d let The () The covrce trx of Q = U, =,,, = G s trx σ σ =, M ( ), ( + ) = V = ( Q,, Q ) (3) ( ) = σ M wth =, =,,, =,, =,,, + (b) rk ( M) = d M = ( / ) M (c) ( ) d = (0, σ ), d γ σ χ R V N M d, σ χ R M R (4) = R ( ) MR,, (5) where d deotes covergece dstrbuto s, χ s the ch-squred dstrbuto wth d degrees of freedo 4

4 Rerk3 Sce rk ( M) =, the covrce trx of G = ( U,, U ) s o-egtve defte Thus, the syptotc ch-squred dstrbuto of the sttstc γ (5) c be obted by tkg + M ( / ) M = Goodess of ft test for the ultvrte orl dstrbuto of dsturbces Let Σ d ˆΣ be defed (6) d (8), respectvely Let the Cholesky decoposto of Σ d ˆΣ be ˆ ˆ ˆ Σ= [ L( Σ)][ L( Σ)], Σ= [ L( Σ)][ L( Σ )], (6) respectvely Let L be the verse of L d let ˆ ε be defed (8) Let ˆ z = [ L( Σ )] ˆ ε, =,,, (z,, z ) Z =, (7) ξ = z z = ( ξ,, ξ ), =,,, ψ = ( ξ,, ξ ) (8) The z re kow s the scled resduls(or spherzed dt), the ut sphere ξ re the proectos of the z s o Theore Let the codtos of le hold Let the trx Z d the -vectors ξ, be defed (7) d (8), respectvely The () The syptotc dstrbuto of z s N(0, I) d z,,z re syptotclly depedet The dstrbuto of Z syptotclly does ot deped o Σ (7) (b) The syptotc dstrbuto ofξ s U ( Ω ) d ξ,, ξ re syptotclly depedet roof By (7) d (8), - - h = x ( XX) x = x ( XX) x 0, Thus, we hve by (), the syptotc dstrbuto of ˆ ε s N(0, Σ), whch we wrte s ˆ ε ~ N(0, Σ ), =,,, By Le(), Thus, ˆ ε ε, L( Σˆ ) L( Σ),, ˆ ε ~ N(0, I Σ ) (9) ˆ [ ( )] ˆ ε [ ( )] ε, z = L Σ z = L Σ, (0) whereε s defed (4) Sce ε N (0, Σ ), by (9) - (0), d Le3, we hve z N(0, I), z N(0, I ) () Thus, the desred results of () s proved By (0)- (), the desred result of (b) s obted Let ξ = ( ξ,, ξ ) be defed (8) d letσ d M re defed (4), respectvely Let Q = ξ, =,,, V = ( Q,, Q ), () = R = V ( ), λ= λε ( ˆ)= R ( σ) MR (3) Rerk4 Cosder the ull hypothess (3), where F 0 deotes the N(0, Σ) dstrbuto wth the preter Σ ukow By Theore, the goodess-of-ft test for F0 c be trslted to the goodess-of-ft test for vlues of λε ( ˆ) (3) ξ U ( Ω ), =,, The ultvrte orlty s reected for lrge Rerk5 Theore dctes tht ξ,, ξ re syptotclly depedet U ( Ω) rdo vectors Hece, the crtcl vlues of the test sttstc λε ( ˆ) c be estted by Mote Crlo sulto wth Σ= I 43

5 The lgorth to pleet the test sttstc The lgorth to copute the test sttstc The lgorth to copute λε ( ˆ) (3) cossts of the followg steps: Copute the vlues of ˆε d ˆΣ (8), respectvely Copute the vlue of Z (7) 3 Copute the vlue of ψ (8) 4 Copute the vlue of V () 5 Copute the vlues of R d λε ( ˆ) (3), respectvely 6 The ultvrte orlty s reected for lrge vlue of λε ( ˆ) The lgorth to estte the crtcl vlues The lgorth to estte the crtcl vlues of λε ( ˆ) cossts of the followg steps: Geerte ε =( ε,, ε ) fro the ultvrte orl dstrbuto N (0, I I) By Le(b), copute ˆ ε = ( ˆ ε,, ˆ ε ) = ( I ) ε, X where X s defed (7) 3 Copute ˆ =[ ˆ ε ] ˆ ε Σ / ( p), z [ ( ˆ )] ˆ = L Σ ε, =,,, where Σ ˆ = [ L( Σˆ )][ L( Σ ˆ )] ( the Cholesky decoposto) 4 Copute 5 Copute 6 Copute ξ = z z = ξ ξ = (,, ),,, Q = = R = V ( ), [ ξ ],,,, = where = (/,,/ ) 7 Copute λ = λε ( ˆ )=[ R ] ( σ) MR, (,, ) V = Q Q whereσ d M re defed (4), respectvely Dog these N tes gves sple of replctes λ,, λ N Let λ(),, λ(n) be the order sttstcs, the crtcl vlues for λ c be estted fro λ (),, λ (N) Coclusos Whe the dstrbuto of the dsturbces ε, () eoys ultvrte orlty, the drecto vectors ξ (8) should be, pproxtely, uforly dstrbuted o the surfce of the ut sphere Ω Bsed o the geerlzed verse of the covrce trxσ M of U ( Ω) (4), the test sttstc λε ( ˆ) (3) s costructed whch possesses syetry By Le7(b), the + Moore-erose verse M = ( / ) M Hece, the proposed test sttstc λε ( ˆ) c be coputed esly for y deso of the dsturbce vector The ellptclly syetrc dstrbuto s turl exteso of the ultvrte orl dstrbuto The dsturbce ultvrte ler odel c be ssued to hve ellptcl dstrbuto robustess studes Bsed o property for the sphercl syetry d the odfed EDF test, Su d Guo(0) suggested the test procedures for testg the ellptcl dstrbuto[7] The goodess-of-ft test for the ultvrte orl dstrbuto of the dsturbces the ultvrte ler regresso odel(mlrm) c be exteded to testg the ellptcl dstrbuto of the dsturbces MLRM 44

6 Ackowledgeets Ths pper s supported by the Fudetl Reserch Fuds for the Cetrl Uverstes Refereces [] Gero MDJ, Grcí JM, Meís R, Testg goodess of ft for the dstrbuto of errors ultvrte ler odels Jourl of ultvrte lyss, 005, 95, 30~3 [] Su Y, Yg ZH, Goodess-of-ft lyss for uforty o the surfce of ut sphere Act Mthetce Applcte Sc, 009, 3 (), 93~05 ( Chese) [3] Su Y, Yg ZH, Goodess-of-ft test for uforty o the surfce of ut sphere bsed o geerlzed verse I : Recet Advce Sttstcs Applcto d Relted Ares (Coferece roceedgs of The 4th Itertol Isttute of Sttstcs & Mgeet Egeerg Syposu, Dl, Ch) Edted by Zhu KL, Zhg H Sydey: Ausso Acdec ublshg House, 0, rt, 33~37 [4] Greee WH, Ecoortrc Alyss, 4th ed retce Hll, Ic, 000 [5] Huffer FW, rk C, A test for ellptcl syetry Jourl of ultvrte lyss, 007, 98, 56~8 [6] Fg KT, Kotz S, Ng KW, Syetrc Multvrte d Relted Dstrbutos Lodo, New York: Chp & Hll, 990 [7] Su Y, Guo LH, A chrcterzto-bsed test for the ellptclly syetrc dstrbuto I: 5 th Itertol Isttute of Sttstcs d Mgeet Egeerg Syposu 0: Dt-Drve Mgeet Scece uder Developg Sydey: Ausso Acdec ublshg House, 0, [8] Mrd KV, Mesures of ultvrte skewess d kurtoss wth pplctos Boetrk, 970, 57, 59~530 [9] Lg JJ, WSY, Yg ZH,Chrcterzto-bsed Q-Q plots for testg ultorlty Sttstcs & robblty Letters, 004, 70, 83~90 [0] Murhed RJ, Aspects of ultvrte sttstcl theory, New York Chchester Brsbe Toroto Sgpore : Joh Wley & Sos, Ic, 98 [] Zhu LX, Zhu RQ, Sog S, Dgostc checkg for ultvrte regresso odels Jourl of Multvrte Alyss, 008, 99, 84~859 [] Díz-Grcí JA, Gutérrez-Jáez R, The dstrbuto of the resdul fro geerl ellptcl ultvrte ler odel Jourl of Multvrte Alyss, 006, 97, 89~84 [3] Ng VM, Robust byes ferece for seegly urelted regressos wth ellptcl errors Jourl of Multvrte Alyss, 00, 83, 409~44 [4] Wtso GS, Sttstcs o spheres, New York : Joh Wley & Sos, Ic, 983 [5] Glberto A ul, Mrco Mederos, Flldor EVlc-Lbr, Ifluece dgostcs for ler odels wth frst-order utoregressve ellptcl errors Sttstcs d probblty letters, 009, 79, [6] yöe S, Dstrbuto of rbtrry ler trsforto of terlly studetzed resduls of ultvrte regresso wth ellptcl errors Jourl of Multvrte Alyss, 0, 07, 40~5 45