A note on testing the covariance matrix for large dimension

Transcription

1 A ote o tetg the covarace matrx for large dmeo Melae Brke Ruhr-Uvertät Bochum Fakultät für Mathematk Bochum, Germay e-mal: melae.brke@ruhr-u-bochum.de Holger ette Ruhr-Uvertät Bochum Fakultät für Mathematk Bochum, Germay e-mal: holger.dette@ruhr-u-bochum.de FAX: ecember 9, 2003 Abtract We coder the problem of tetg hypothee regardg the covarace matrx of multvarate ormal data, f the ample ze ad dmeo atfy / = y. Recetly, lm, everal tet have bee propoed the cae, where the ample ze ad dmeo are of the ame order, that y 0,. I th paper we coder the cae y =0ady =. It demotrated that tadard techque are ot applcable to deal wth thee cae. A ew techque troduced, whch of t ow teret, ad ued to derve the aymptotc dtrbuto of the tet tattc the extreme cae y =0ady =. AMS Subject Clafcato 62H5; 6220 Keyword ad phrae: phercty tet, radom matrce, Whart dtrbuto Itroducto A commo aumpto multvarate tattc that of a..d. ample of ormally dtrbuted -dmeoal radom varable X,...,X + Nµ, Σ, where µ R deote the mea vector ad Σ R a potve defte covarace matrx. I th paper we are tereted the problem of tetg the hypothe of phercty, that. H 0 : σ IR >0 : Σ = σi, H : Σ σi σ IR >0, ad the problem of tetg the hypothe.2 H 0 :Σ=I, H :Σ I,

2 f the ample ze ad dmeo coverge to fty, where I R deote the detty matrx. Note that the detty matrx I.2 ca be replaced by ay other potve defte matrx Σ 0 R by multplyg the data by Σ /2 0. Clacal aaly aume that the dmeo fxed, whle the ample ze coverge to fty. However, recet applcato gee expre expermet requre tattcal method, whch are applcable the cae, where the dmeo large compared to the ample ze ee Sebat, Guo, Kohae ad Ramo 2003]. Becaue lkelhood rato tet ee e.g. Murhead 982 or Adero 984] are ot applcable the cae, Ledot ad Wolf 2002 uggeted to ue certa alteratve tet whch were orgally propoed by Joh 97 ad Nagao 973 the cae, where the dmeo fxed ad the ample ze coverge to fty. To be prece, let X =X,...,X + ad defe.3 S = XXT X X T a the ample covarace matrx, where X = + + j= X j the mea of the obervato. For tetg the hypothe of phercty Joh 97 propoed to reject H 0 for large value of the tattc U = 2 tr S.4 / tr S I where tr A deote the trace of the matrx A. For the problem of tetg the hypothe H 0 that the covarace matrx gve by the detty Nagao 973 propoed to reject the hypothe.2 for large value of the tattc.5 V = tr S I2, Recetly, Ledot ad Wolf 2002 vetgated the properte of thee tet f the ample ze ad the dmeo coverge to fty at the ame rate, that.6 y 0,. They proved aymptotc ormalty for a approprately tadardzed vero of U ad V ad howed that th cae the tet of phercty baed o the tattc U tll cotet, whle the tet for the hypothe H 0, whch reject the ull hypothe for large value of the tattc V, ot ecearly cotet uder aumpto.6. For th reao Ledot ad Wolf 2002 propoed to reject the ull hypothe H 0.2 for large value of the tattc W = tr S I tr S + ad howed aymptotc ormalty of a tadardzed vero of W ad cotecy of the correpodg tet. The proof cot of a applcato of the delta-method to the vector tr S, tr S2, for whch the aymptotc ormalty ha bee etablhed by Joo 982. I the preet ote we vetgate the properte of the tattc U, V ad W the extreme cae where the ample ze ad dmeo are ot of the ame order, that.8 y {0, }. 2

3 We wll demotrate that thee cae approprately tadardzed vero of the tattc U, V ad W are tll aymptotcally ormal dtrbuted. However, the method ued by Ledot ad Wolf 2002 to prove thee reult uder aumpto.6 ot applcable ay more. Thee dffculte are llutrated Secto 2. I Secto 3 we ue a drect argumet to etablh aymptotc ormalty uder the aumpto.7, whch of t ow teret ad baed o pecal tructure of the Whart matrx. Th techque avod the applcato of the delta-method ad may be ueful to vetgate other properte of the ample covarace matrx. Fally, ome of the more techcal argumet are gve the Appedx ee Secto 4]. 2 The proof baed o the delta-method revted I th ecto we brefly expla the dffculte the applcato of the delta-method to etablh aymptotc ormalty of the tattc U, V ad W uder aumpto.8. The varace of trs ad tr S2 ca be calculated by tadard method, that Var tr S = 2 Var trs2 = efe F = tr S tr S] 2. tr S2 tr S2 ], the the followg reult ca ether be etablhed by mlar method a gve Joo 982 or by the ame argumet a gve the proof of the reult Secto 3. Lemma 2.. If aumpto.8 atfed wth y =0, the F N 0,A where the aymptotc covarace matrx A defed by 24 A =. 48 If aumpto.8 atfed wth y =, the varace of both compoet of the vector F are of dfferet order. It follow from Lemma 2. that the delta-method ot applcable the cae y =, becaue we caot tadardze both compoet of F multaeouly order to obta a odegeerate lmt dtrbuto. Note that the frt compoet tr S requre a tadardzato wth order to obta a potve lmtg varace. However, uch a tadardzato would yeld for the varace of the ecod compoet Var tr S2 = o 2 2,

4 whch coverge to fty. Moreover, eve the cae y = 0 the applcato of the delta-method wth the aymptotc ormalty Lemma 2. produce ubtatal dffculte. For th we coder examplarly the tattc U whch obtaed a U = g tr S, tr S2, where the fucto g defed by gx, y =y/x 2. Obervg that ] ] tr S =, tr S2 = + + ee Joo 982] we obta that the radom varable U g, + + aymptotcally ormal dtrbuted wth mea 0 ad varace , + 2 = = o. 48 Coequetly, the tadardzato from the frt part of Lemma 2. yeld a degeerate lmt dtrbuto for the tattc U + the cae where / 0. Moreover, the ame dffculte appear f th techque ued to derve the aymptotc dtrbuto of the tattc V ad W. I the followg ecto we ue a alteratve methodology to derve aymptotc properte of thee tattc. 3 The Whart matrx Note that uder the ull hypothe H 0 or H 0 the etmator S.3 ha eetally up to a calg factor a -dmeoal Whart dtrbuto wth degree of freedom. The followg Lemma how that tr S ad tr S 2 have rather mple repreetato term of χ 2 -dtrbuted radom varable. Throughout th paper the ymbol = mea equalty dtrbuto. Lemma 3.. If W deote a -dmeoal Whart matrx wth degree of freedom ad covarace matrx Σ=I,the tr W = m{,} = Y + X + tr W 2 = m{,} 2 = 2 Y + X m{,} 2 4 = Y X ++

5 where X l χ 2 l ad Y χ 2 are depedet ch-quare dtrbuted radom varable. A a applcato of Lemma 2. we ca fd very mple expreo for the tattc U, V, W term of depedet radom varable wth a χ 2 dtrbuto. Thee reult ca be ued to obta a drect proof of aymptotc ormalty baed o a cetral lmt theorem for m-depedet tragular array of radom varable. We wll llutrate the geeral trategy for the tattc V the cae, all other cae are treated mlary. The followg lemma gve a aymptotc equvalet repreetato for the tattc V. The proof traghtforward ad therefore omtted. Lemma 3.2. If the ull hypothe.2 vald ad lm / = y, ] the tattc V, defed.5 atfe 3 V + = Y, + A. Here A = o p ad the radom varable Y, are gve by ] Y, = wth = Y + X +, = ] = 2 +2Y X ++, where X,Y are defed Lemma 3.. Theorem 3.3. If the ull hypothe.2 vald ad where / := 0. 3 V + lm / = y, ], the, N 0, 4 y +8, The followg theorem cota the correpodg tatemet for the cae lm / = y 0,, ad for the tattc U ad W defed by.5 ad.7, repectvely. The proof are mlar ad therefore omtted. Theorem 3.4. If the ull hypothe.2 vald ad V + N 0, 4+8y lm / = y 0, ], the, Remark 3.5. It wa proved by Nagao 973 that the tet, whch reject the hypothe H 0 :Σ=I 5

6 for large value of the tattc V, cotet f the dmeo fxed ad the ample ze coverge to fty. Ledot ad Wolf 2002 demotrated that th tatemet ot correct f ad coverge at the ame rate to fty. Theorem 3.3 how that the tet tll cotet f lm / =0., The followg reult how that the tet baed o W for the hypothe H 0 ad the tet baed o U for the hypothe of phercty alway cotet depedetly of the value y = lm / 0, ]., Theorem 3.6. If the ull hypothe.2 vald ad W lm / = y 0, ] the, N0, 4 Theorem 3.7. If the ull hypothe. of phercty vald ad U + N 0, 4 lm / = y 0, ], the, Ackowledgemet. The work of H. ette wa upported by the eutche Forchuggemechaft SFB 475, Komplextätredukto multvarate atetrukture. The author would lke to thak Iolde Gottchlch who typed umerou vero of the paper wth coderable techcal experte. 4 Appedx: Proof Proof of Lemma 3.. We wll oly coder the cae, the remag cae ca be treated by mlar argumet. By t defto a -dmeoal Whart matrx wth degree of freedom ad covarace matrx Σ = I ca be repreeted a 4. W = YYT where Y =Y j j=,..., =,..., a matrx wth depedet detcally dtrbuted etre, uch that Y j N0,. It ow follow from Slverte 985 that the matrx W orthogoally mlar to a tragular matrx à =ã j,j=,..., wth etre ã, = 2 Ỹ + + X + 2, =,..., ã,+ = Ỹ X +, =,..., ã +, = Ỹ X +, =,...,, 6

7 where Ỹ =0, X 2 χ 2 ; Ỹ 2 χ 2 are depedet χ 2 dtrbuted radom varable X 0; Ỹ 0. Therefore t eay to ee that the matrx W ha the ame egevalue a the tragular matrx A =a j,j=,..., wth etre Ỹ 2 a, =ã +, + = + + 2, =,..., a,+ =ã, + = X ++, =,..., Ỹ a +, =ã +, = Ỹ X ++, =,..., Itroducg the otato X = X 2,Y = Ỹ 2 the aerto of the Lemma ow obvou. Proof of Lemma 3.2. By Lemma 3. we have 3 V + = = ] ] Y 2 ] Y Y X Y X ] 3 Y X + Y X + ] +2 Y 3 Y ]. ad a traghtforward but tedou calculato ow how that the lat four term are of order o p ee Brke 2003]. Proof of Theorem 3.3. By Lemma 3.2 we obta the repreetato 3 V + = Y, + A, where A = o p ad the radom varable Y, are defed 3.. Note that the radom varable {Y, } =,..., form a tragular array of rowwe m-depedet radom varable. Moreover, Y, ]=0, ad a traghtforward but tedou calculato yeld = 4.2 lm Var Y, = 4, y +8 = ee the remark at the ed of th Appedx. A cetral lmt theorem for m-depedet tragular array wa gve by Orey 958, but we foud the codto of a cetral lmt theorem for 7

8 tragular array of α-mxg radom varable eaer to check. To be prece we appled Theorem 2. of Lebcher 996 to the equece For th we calculate Y, = ] Y, 4 = N. ] 4 ] 3 ] 2 ] ] 4 ] ] 3 ] 2 ad ] Y, 6 = ] 6 ] 5 ] 4 ] 3 ] 2 ] ] 6. ] ] 2 ] 3 ] 4 ] 5 Becaue the dtrbuto of the radom varable ad explctly kow we ca evaluate = Y, k k =4, 6 explctly ee Brke 2003 for more detal]. I the followg we deote by p j, k adq j, k polyomal of degree k N ad argumet, the we obta by a tedou calculato = ] Y, 4 p, 6 = + p 2, 5 + p 3, 4 + p 4, p 5, 2 + p 6, p, 0 + = O 2 8,

9 = = ] Y, 6 q, 9 = + q 2, 8 + q 3, 7 + q 4, 6 + q 5, q 6, 4 + q 7, 3 + q 8, 2 + q 9, q 0, 0 2 = o, Coequetly t follow from Theorem 2. ad equato 2. Lebcher 996 ] ] N0, 4 y +8, whch prove the aerto of Theorem 2.. Referece T.W. Adero 984. A Itroducto to Multvarate Stattcal Aaly. Wley, N.Y. M. Brke Zufällge Matrze be multvarater Normalvertelug. plomarbet, Fakultät für Mathematk, Ruhr-Uvertät Bochum Germa. S. Joh 97. Some optmal multvarate tet. Bometrka, 58: S. Joh 972. The dtrbuto of a tattc ued for tetg phercty of ormal dtrbuto. Bometrka, 59: 69/73.. Joo 982. Some lmt theorem for the egevalue of a ample covarace matrx. J. Multvarate Aal., 2: -38. O. Ledot ad M. Wolf Some hypothe tet for the covarace matrx whe the dmeo large compared to the ample ze. A. Statt., 30 4: Lebcher 996. Cetral lmt theorem for um of α-mxg radom varable. Stochatc ad Stochatc Report, 59: R.J. Murhead 982. Apect of Multvarate Stattcal Theory. Wley, New York. H. Nagao 973. O ome tet crtera for covarace matrx. A. Statt., : S. Orey 958. A cetral lmt theorem for m-depedet radom varable. uke Math. J., 25: P. Sebat,. Guo, I.S. Kohae, M.F. Ramo Stattcal challege fuctoal geomc. Stattcal Scece 8, J.W. Slverte 985. The mallet egevalue of a large dmeoal Whart matrx. A. Probab., 3: