･孟皇 J a b e f J c d ･ i 主JbcdeJaf I ( 一志 + 孟寺) 皇 J a e f J b c d

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EconomicReview(OtaruUniversityofCommerce),

Vo159,No.2&3,95‑107,December,2008.

Errataandsupplementtothepaper

"High er‑orderasymptoticcumulants

ofStudentizedestimatorsincovariancestructuresM

HaruhikoOgasawara

ThisnoteistosupplementOgasawara(2008)withsomeerrata.Inthis note

El(mabcd‑E(mabcd)Hmefgh‑E(mefgh));n‑I] El(mabcd‑E(mabcd))(sef‑qef)(sgh‑C,gh);nL2] andE((mabcd‑Jabcd)(sef‑Jef)(sgh‑Ugh);n12) w

i llbegiveninLemrnas3,4and5,respectively.Thesamplemomentmabcd wasdefined in(2.6)whilesob istheusualunbiased samplevariancei.e.,

∑l"=.(^X,â^‑^Xa⁾⁽^x,^b‑Xb^)/⁽^N‑1^).^si^ncê^X^M‑^Xa‑J^X:â‑F(^Xâ⁾^{‑氏 ‑E}⁽^ka⁾⁾^,

Xt.a andXa areredeflned in thisappendixasthedeviationsfrom their expectations.LetSa.020k‑∑ ,"=.X,a.X,arX,ak(k‑1,2,‑),then

mabcd‑去 soぉd‑志士 sasぉd･L 皇 sasbScd一言 sasbScSd,

1

sef‑去 sef一両 SeSf･

ThefわllowingLemmas1and2willbeusedinLemmas3and4.

Lemma1. 3

E(makd)‑^{去 (}^N²^‑3N^･3⁾⁽^N‑¹⁾^Jobcd･^{L (}²^N‑3)⁽^N‑1)∑^J^a^b^Jc^d

=1111三二+

(A.1)

鉢おd･(i一言 )i^Jab^J cd･0(N‑3,･ 'A'2'

U N N2ノ〉^abcd LN N2)L^〉ab^〉 cd'〉＼ーT ''

proof.NotingthatX,a,Saredeviationsfrom theirexpectations,from (A.1)we have(A.2).Q.E.D.

〔95〕

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96 商学討究第59巻第2･3号

Lemma1canalsobeobtainedbygeneralizingtheunivariateexactresult givenbymathStatica(RoseandSmith,2002):

E(mMaa)‑去 (N2‑3N･3)(N‑1)JMaa･^{i (}^2N‑3⁾⁽^N^‑^.⁾^Ji.⁽^A.³^'

Lemma2.

E(m abcdSef,‑(去 ‑芸 )^Ja^khf･(I‑吉 .# )Jabc^dJef･孟皇 Jab^efJcd

･右転 hUafI(一志 +錘 qae^fJbcdI(i 一錘 ^Jâ^b^Jc^d^Je^f⁽Â^.^., 一志皇(JaeJbf仰 ê⁾^J^c^d.0(^N^13)･

proof.From(A.I),takingtermsuptoorder0(n‑2), 1

E(mabcdSef)‑両 (NJabcdefI(" 2‑N)gabcdJefi

1 4

一両 (4"Jabcdef'(" 2‑")4gabcdJefI(N2‑")∑ qaefUbcd)

I 6 4

'両 (("2‑")(6gabcdJ ef ' 2∑ gabefgcd'3∑gaefgbcd) 3

+N(N‑1)(NI2)2C,ef∑ C,abOIcd)

3

一志 N(N‑.,(N‑2,Jef^{∑ Ja}^b^Uc^d

1

一両 ("uabcdefI(N²‑N)Jabcdqefi

1 8 4

'両 (" 2‑")(4gabcdJef'∑ Jbc^hJ4'∑ 6‑^fJbcd)

1 ³ 6

一両 "("‑1)("‑2)(2qef^∑ ^qa^b^qc^d

'

^{∑ (}^qâê^gb^f+Ja^f^qbê^{叛 d}⁾

+o(〃3)

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Errataandsupplementtothepaper"HlghemrderasymptodccumuLantsofStudendzedestlmatOrSlnCOVananCeStruCtureS 97

去 (.一芸一志)J a b c d ef I(1一芸.宗一去+芸 )J^abcdqef

･孟皇 J a b e f J c d ･ i ^主

^Jb^c^d^e^Ja

^f Î ⁽ ^一 ^志 ⁺ ^孟寺) ^皇 ^J â ê ^f ^J ^b ^c ^d

I ( 堅 ^欝 ‑ 嘉一 i) q e f 主 g a b q c d ‑ i 皇 ( q a e g b f ･ q a f

^gb^e^{h d '}^A'⁵^'

+o(〃³⁾,

whichgives(A.4).Q.E.D.

Then,wehave Lemma3.

El(m abcd‑ E (m abcd)H m efgh‑ E(m efgh))]

4

‑去(Jabcdefgh‑∑(qefghaqkd.Jabcdeqfgh)‑JabcdUefgh

42=16

･∑ o･bcdC,fghqae)+0(N2)･

(A.6)

ProorFrom (A.1)andLemma1,

‑E(mobcd)E(mefgh)

‑‑((.‑;)uabcd･

志

^士^gab^gc^dH(1‑;⁾^ge^f^g^h^.諾 ^ge^fUg^h⁾^I0(^N‑²^,

‑ ( I . . 芸 ) U

â^k^dÛê^f^g^h

‑ 孟 U d c d i u e f g g h 一･孟 J e / g h 皇 q a b J

^c^dI0(^N‑²^,^,

E(maぉdmefgh,‑iJak^h^f^g^h一諾 (^‰ JbcdIJaぉhJfgh,I(I‑^; ⁾^Ja^b^c^dJe^f^g^h

･諾

^qb^c^dJ

/

^g

h q a e I i

ⁱ⁽^Ja^b^c^dJe^f^JghI

J e / g h U a b J c d

^{'･ 0 'N ‑2}^'^,

(A.7) whichgives(A.6).Q.E.D.

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.()LLil 商学討究第59巻第2 ･3号 Lemma4.

El(mabcd‑ E(mabcd)Xsef‑JefXsgh‑Ugh)]

‑三 ^C'^a^b^c^h/gh‑(qabcddO'gh+ C'^ab^c^d^ghq^d)

4

‑∑ (JbcdefJagh+C,bcdghJa〆+C,aefghJbcd)

4 4

‑∑^Jakde^qf^{gh‑ Ja}^ぉdJe^f^ghI2^C,â^b^c^dJdqgh‑∑ (^Jaê^fqghI^C,â^ghOlq)^0‑^b^c^d

4

+∑ (^JagJe,A‑^{W a}^hC^,ê^f^g十^gaêqgh^f+^C,â^fJghê)^Jb^c^d

4C2=6

+^{∑ i}⁽^Ja^d^･^qb^g^h+^Ja^g^hJb^e^f)qcdI(0‑udC,bgh+^Ja^ghJb^c^d)old･

+(qacdJbef+Jmfqbcd)Ugh)]+o⁽^Ⅳー³⁾^.

Proof.Decomposetheexpectationin(A.8)as

E(mabcdSefSgh)‑ E(mabcd)E ((sef‑qefXsgh‑C'gh))‑ E(mabcdSef)cTgh

‑E(mabcdSgh)Jef+E(mabcd)qefggh･

WederivetheaboveexpectationsonebyoneinthefわllowingSectionsatod･

a.E(mabcdSdSgh)

Since

mabcd

m‑ S〆Sか‑議等 sqsか ‑(N̲1XN²̲N)S^dS^gSh

(N̲.XN2̲N^,S^かSêSf･務 sês^fS^gSh,^̀Â●¹⁰^'

mabcd

wetaketheexpectatiorLSOfthefourtermsontheright‑handsideof(A.10)in t

hefollowingSubsectionsa.1toa.4.

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Errataandsupplementtothepaper"HlghemrderasymptodccumuLantsofStudendzedestlmatOrSlnCOVananCeStructures

a.1E(m abcdSefSgh/(N‑1)2)

E(苧 sefS^gh)

両 (NJobchfgh･(N2IN)(JakhfJghIJabcdghJef

･6abcdJefgh'･"'"‑1''"12'6akdJef.Jgh'一義可 [̀" 2‑q

x4(gabcLkfggh+C,abcdghC,ef)I(N²‑N)∑ (Jbc.kfgagh+C,bcdghJaefIJaefghJbcd)

+(N2‑N)4JabcdJefgh

十"'"‑1''" I 2'{460bcdJefJgh･玩 fJkdJghIJaghJ?dJef'']

･ _･

^空 ^車 ^[^"^'^"‑¹^'^̀^"‑2^'^{{ 660}^b^c^d^Je^fJg^hIJe^f^g^h^{2E Ua}^b^Jc^d

∑ (C,aefJbghJcdIJaghC,befC,cd

+0‑acdO‑befJgh+0‑acdJbghJef+0‑bcdJoefC'gh+0‑bcdO‑aghJef) 6

･2∑ (JchfJabC,ghIgcdghJabJef) )

･"'":l''"‑2''"‑3'gefUgh2igab

g c d ]

^,

一雨 "("^I1)("^‑2)("‑3)3gefJgh∑ 6abgcd+0(N13)

99

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100 商学討究第59巻第2 ･3号

‑去 Jabcdefgh.(嘉一i )(JabchfJghIgaぉ dghJef,

‑宗主(^JkhfJa^gh^.J^b^c^d^ghJa^e^f

･JaefghUbc d ,I(孟一芸 )Jo^b^c^d^Jefgh

･(崇一芸㌔ 4･荒 ₆₎^(㌔ ^b^c^d^Je^f^{C T}^gh N‑2 ーti, N‑2 ̲I

2∑(crcdyJabCrgh.gcdghJabJef) jY‑∠‑ 1YT∠■

･両 Jefgh2∑ JabJcd･両

‑荒

土(

^6‑^fgkd^Jg^h･JaghJbcdJef,

･^荒土(JaefJb^g^hJc^d^I^Ja^g^h^Jk^f^Jc^d

+C'acdO‑befJgh+ JacdC'bghCref+CTbcdCraefCr^g^h+^C'^b^c^d^CT^a^g^hCT^e^f)

(〟‑2)(〟‑3)

Ⅳ2(〟‑1) Ⅳ3(〟‑1)₃_)Je0‑e_ffJJgghh≦∑ JabC,cdI0(N‑3)

‑去 J abchfgh･(去 ‑i )(JakdefJghIgabcdghqef,

I 去

^去 ⁽^6b^c^d^e^f^qa^ghI

^u ^b ^c

^dgh^ga^f^I^Ja^e^f^gh^Jb^c^d^,

･(去一鉢 bcdJ efghI(.1孟 +; )JabcdJef^g^{g h}

2 3

･

^# ^e^f^g^h^{∑ qa}^b^Jc^{d･宗主 (}^Jc^h^f^qa^b^Jg^hIJc^d^g^h^qa^b^Ue^f,

･(一志 .去)皇(^Ja^e^f^Jb^c^dJghIJaghJbcdJef,

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･宗^{主 (}^{64 gb}^g^hJc^{d･ Ja}^g^h^Jb^e^f^Jc^d

+cTacdCTbefJgh+C'acdC'bghJef+JbcdO‑ufo‑gh+0‑bcdCTaghJef)

I^{(孟 ‑}A )JefJgh主^Jab^Jcd･0(N13,･

a.2‑E lm abcdSefSgSh/((NI1)(N2‑N))]

Ⅳ 2(〟‑1)2(N(N‑1)(gabcdefC'gh+Jobcdgh0‑ef)

+N(N‑1)(JabcdgJejh+^uabcdhO‑erg)+N(N‑1)JabcdJefgh

+N(N‑1)(N‑2)JabcdJefJgh)

Ⅳ3(〟‑1)2〟(〟‑1)(〟‑2)4^cT^ab^c^dCTefCTgh

･^{∑ (}4^Jb^c^d^g^C^,ô^hô^‑ê^fI^C^,^b^c^d^h^Ja^g^C^,ê^f)

･皇

( J a g J

epIJahJefgIJoefJghIJefJagh,Jbcd)

Ⅳ 4(^〟‑1)2〟(〟‑1)(〟‑2)(〟‑3)

4C2‑6

･

^{∑ (}JagJbh‑WahJbg.qabJgh)JcdJefI 0 (N 3)

101

(A.ll)

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102 商学討究第59巻第2 ･3号

一志 (JobcdefJghIJaぉdghJef)一志 (JabcdgJep.JabcdhUefg)

1

京 g^abcdg^e^fgh

I(一志.i )JobcdJefJgh定 ^立 (JbcdgJohJef･^J ^bcdhJagJef,

定立 (^Ja^gJep.^JohJefgI^Ja^e^fJgh.^Je^fJa^g^h,Jkd

一志

^{皇 (}^Ja^gJb^{h‑ a}^hJb^g^I^Ja^bJgh^,^Jc^dJe^f^I0(^N‑^3,･

(A.12) a.3‑ElmabcdSgh^SeSf/((N‑1)(N2IN))]

Theresultcanbeobtainedfrom Subsectiona.2byexchangingsubscriptse andfwithgandh,respectively.

a.4 E (m abcdSeSfSgSh/(N2‑ N )2)

E〈f " sh⁾

雨 "("ll)(N‑2)(JefJgh･uegupIJehUfg)JabcdI0(N3)

‑去 (JefJghIJegJpIJehJfg)Jabcd･0(N‑3).

(A.13) FromSubsectionsa.1througha.4with(A.10),wehave

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ÈmabcdSefSgh'‑i qabcdefgh+(去‑i⁾⁽^Ja^b^c^d^e^{f? h}IJabcdghJef'

‑ 去(

î:(ûkdê^fJa^gh^十^gb^c^d^g^hJ‑IÎ6^‑^f^g^h^Jb^c^d,I^{∑ qa}^b^c^hJf^g^{h )}

･(去一鉢 bcdJefghI( l一三･鈷 bcdJefUghIi Jefg h主 JabJcd

･鈷 bcd(JegJ^{p ･ J}^e^h^Jf^g,I2i^{(J c}hfggh^{I J}^c^d^g^h^Je^f^{)J ob}

･^{∑ i}4⁽^c^,^b^c^d^gC,ô^hÎ^Jkd^h^ga^g)â‑ê^f

I(Jk&JafIJ^kqJ^ae,Ugh, ].(‑去･錘 (Jaef^Jg^hI^{㌔ hJe}^f,Jkd

･去i(Jag^{q e}^p ^.^qa^h^Je^f^gÎûaêû^g^h^fI^Jâ^f^J^g^k^,ûkd

･宗主"Jâê^f^J^b^g^hI^Ja^g^h^Jbê^f⁾^Jc^dI(^Jâ^c^d^J^b^g^h^･Ja^g^h^Jb^c^d^,^Je^f

I(JacdJbefIJaefgbcd,Ugh,I(喜一i )JefUgh主 JabJcd

一志皇i(JagU^bh^{I Ja}^hUb^g)Ue^f^I(JMJbfI uafUk,qghiJcdI0(N‑3)･ b.‑E(mabcd)E ((Ref‑C'efXsgh‑Ugh))

‑E (m obcd)E i(sef ‑C'ef^Xsgh‑Ugh))=

( 一志

^･^{A )Ja}^b^c^d^Je^f^g^h

103

(A.14)

I (

去一鉢 bcdUefUgh‑i U abcd(Jeg6p‑ ehJfg,一志 6efghi JabJcd

2 3

'前 gefUgh∑ JabJcd'0(N13),

(A.15) where

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104 商学討究第59巻第2 ･3号 1

E{(sef‑JefXsgh‑Ugh,}‑去 (Jefgh‑Jefqgh,.両 (JegJpIJehJ/g,

‑^{去 (}^Je^,^g^h‑Je^fJgh)･去(qegJp^+Je^hJ/g)I0(N13) (seee.g.,Kaplan,1952)isused.

C.‑E(mabcdSef)0‑gh‑E(mabcdSgh)Jef

‑E(mabcdSef)crgh‑ E(mabcdSgh)clef

‑(‑;･i )

(

(JabcdefJgh+crabcdghO‑ef)

10 20

+Ll2'嘉示 ₎_Ja_b_c_dge_f_qg_h一志皇 (JabefJghIJabghgef,qcd

一志主(gbcゐgafggh+gbcdggahUef)

I(去‑i )ⁱ ^'^gâê^fg^:^c^dÛ^g^hI^Jâ^g^h^Jb^c^d^Jê^{f '}

I(一芸･#)JefJgh∑ ^{g abJ}cd･古土{(W bf･ uafJbe,Ugh

+(ologJbh+ cTahJbg)Jef)oIcd･

d.E(mabcd)Je/ 'gh

E(ma bcd^,Jef^J^gh‑(1⁺ ^鈷 ^c^{d g}^e^f^J^gh

･^{(孟 ‑ i} )^qef^Ughⁱ^{J a}^b^q^c^dI0(^N^‑³^,^･

Summing(A.14),(A.15),(A.17)and(A.18),(A.8)follows･QIE･D･

(A.16)

(A.17)

(A.18)

･ 孟皇 J a b e f J c d ･ i 主JbcdeJaf I ( 一 志 + 孟寺) 皇 J a e f J b c d

'

･ 孟皇 J a b e f J c d ･ i 主

f I ( 一 志 + 孟寺) 皇 J a e f J b c d

I ( 堅 欝 ‑ 嘉一 i) q e f 主 g a b q c d ‑ i 皇 ( q a e g b f ･ q a f

志