ASYMPTOTIC APPROXIMATIONS FOR DISTRIBUTIONS OF TEST STATISTICS OF PROFILE HYPOTHESES FOR SEVERAL GROUPS UNDER NON-NORMALITY

Transcription

1 Joural of ahemaal Sees: Avaes a Alaos Volume Number Pages 95-6 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS OF TEST STATISTICS OF PROFILE HYPOTHESES FOR SEVERAL GROUPS UNDER NON-NORALITY YOSIHITO ARUYAA Dearme of ahemaal Sees Osaa Prefeure Uversy - Gaue Naa-u Saa-Cy Osaa Jaa Absra The urose of hs aer s o suy he srbuos of sass for esg some hyoheses oerg omarso of rofles The hyoheses osere lue he arallelsm hyohess he level hyohess a he flaess hyohess We erve asymo exasos of he srbuo fuos of es sass for hese hyoheses a hee of he uer ereage os of he sass u o he frs orer erm wh rese o he verse of he samle sze uer geeral srbuos Irouo Suose wo eee grous or samles reeve he same se of ess or measuremes If hese ess are omarable for examle all o a sale of o he varables wll ofe be ommesurae The he oulao mea veors µ ( µ L µ ) a µ ( µ L µ ) ahemas Subje Classfao: 6H 6E 6H5 Keywors a hrases: asymo srbuo asymo exaso Hoellg ye sas o-ormal oulao Reeve July 9 Sef Avaes Publshers

2 96 YOSIHITO ARUYAA where he symbol ' ' aes he rasose oerao are he mea rofles of wo grous Raher ha esg he hyohess ha µ µ we wsh o be more sef omarg he rofles obae by oeg he os ( µ ) L a ( µ ) L The es for smlary of rofles was gve by Rao [7] a a wosamle geeralzao of oe-samle resul was susse Greehouse a Gesser [6] There are hree hyoheses of eres omarg he rofles of wo samles The frs of hese hyoheses aresses he queso Are he wo rofles smlar aearae? or more resely Are hey arallel? If he wo rofles are arallel he oe grou sore uformly beer ha he oher grou o all ess The seo hyohess of eres omarg wo rofles s Are he wo oulaos or grous a he same level? Ths hyohess orresos o a grou (oulao) ma effe he uvarae aalyss of varae aalogy The hr hyohess of eres orresog o he es (or varable) ma effe s Are he rofle fla? Noe ha he es sass for hese hyoheses are esseally Hoellg ye sas ormal oulaos The srbuos of Hoellg ye sass have bee wely sue Uer ormal assumo asymo exasos of he srbuo of Hoellg s geeralze rero (Lawley-Hoellg sas) have bee reae by Soa ([9] []) a Io ([9] []) I geeral oulaos s fful o heoreally oba he exa ull a o-ull srbuos However asymo exasos for he srbuos of es sass a be erve u o he orer as he sze of samle es o fy Iwasha [] has obae a asymo exaso of he srbuo of Hoellg s rero whe uerlyg srbuo s ellal The relae sussos geeral srbuos have bee gve by Kao [] Fujosh [] a Yaaghara [] aruyama [5] has reae asymo exasos for srbuos of es sass wo-samle rofle aalyss uer he o-ormal assumo The aalyss of rofles for several oulaos roees muh he same fasho as ha for wo oulaos I fa he geeral measures of omarso are aalogous o hose jus susse by orrso [6] a Srvasava [] I hs aer we exe he wo-

3 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS 97 samle rofle ess o several grous I orer o aheve our urose asymo exasos for srbuos of es sass for rofles from several samles are erve u o orer O ( ) uer geeral srbuos wh fe eghh orer momes Profle Aalyss for Several Grous Through he geeral resuls for he mulvarae lear hyohess we ow exe he rofle aalyss ehque of he wo-samle o several grous We assume ha eee raom samles are obae from -varae oulaos wh equal ovarae mares as he followg layou for he oe-way mulvarae aalyss of varae Poulao grou x x mea µ µ ovarae oal Samle sze observaos x L x x L x overall mea x x xall For oveee we shall wre : The mea of he -h samle s efe as x x j j : a he overall mea s x : x x j j We have he wh (resual) sum of squares a ross-rous marx exresse as E : ( xj x ) ( xj x ) ( ) S j where S s he samle ovarae marx for he -h samle Ths marx s a geeralzao of he oole samle ovarae marx eouere he wo-samle ase The bewee (reame) sum of squares a ross-rous marx s efe as all

4 98 YOSIHITO ARUYAA H : all all The moel for eah observao veor s ( x x ) ( x x ) x j µ ε L ; j L j The veor of resuals ε j of he j-h samlg u has he mulvarae srbuo wh ull mea veor a some uow o-sgular ovarae marx The resual varaes of ffere us are eeely srbue To es equaly of mea veors H : µ L µ we use he usual E a H mares If he varables are ommesurae we a be more sef a exe H o a examao of he rofles obae by log he values µ L µ eah µ We are erese hree hyoheses H ( ): The rofles are arallel H ( ): The rofles are all a he same level H ( ): The rofles are fla The hyohess of arallelsm for wo grous s exresse as H ( ): Cµ Cµ where C s ay (( ) ) marx of ra suh ha C a : ( L ) s he ( ) veor for examle C : L L L To smlfy he oao he subsr wll be roe whe he meso of he veor s lear from he oex The he es sas for esg he hyohess H ( ) s gve by ( ) ( ( ))( ) ( ) ( ): T C x x CE C C x x

5 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS 99 For grous he aalogous hyohess of arallelsm s Ths hyohess s equvale o H ( ) : Cµ L Cµ : µ L µ H a oe-way mulvarae aalyss of varae o he rasforme varables x Cx j By a bas roery j [ x ] C C j x s srbue as [ x j ] Cµ j : E a Cov The wh a bewee mares for esg H ( ) are CE C a CH C We hus use T () ( ) : ( ) rae( CE C) CH C whh s also alle Hoellg s geeralze rero The hyohess ha wo rofles are a he same level s H ( ) : µ µ whh geeralzes mmeaely o rofles a he same level H ( ) : µ L µ For wo grous we use a uvarae -srbuo o es H ( ) as efe by ( ): ( ) T x x E Smlarly for grous we a emloy a F-es for oe-way uvarae aalyss of varae omarg grous wh observaos x j We ulze Wls rero Λ : E ( E H) Ths s of ourse equvale o he F-es o x j a H ( ) a be ese by

6 YOSIHITO ARUYAA T( ) ( )( Λ) ( ) H : () ( ) Λ ( ) E I ormal oulaos T ( ) s aorg o F where F a b s a raom varable of he F-srbuo wh a a b egrees of freeom The hr hyohess ha of flaess esseally saes ha he average of he grou meas s he same for eah varable H ( ) : Cµ () or H ( ) : µ L µ The flaess hyohess a also be sae as he meas of all varables eah grou are he same or µ L µ L Ths a be exresse as H ( ) : Cµ L Cµ The gra mea veor µ () s esmae as Seo by he overall mea x all a H ( ) a be ese by I ormal oulaos ( ) x T () ( ) : ( ) ( Cxall ) ( CE C) C all T s aorg o ( ) ( ) ( ) F Noe ha he arallelsm hyohess H ( ) a he flaess hyohess H ( ) o o ee o he hoe of C as log as C Auxlary Resuls I hs seo we esrbe oaos efos a bas roeres o seal mares For more eals a exelle referee s he boo by Seber [8] Dre rou If A ( ) s a ( q) a a j B s a ( ) s marx he he re

7 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS rou of A a B s efe by he ( q) s marx A B ab : a B L O L a qb a qb The erms Kroeer rou a esor rou are also use he leraure C Le B a D be ( s ) ( q ) a ( ) resevely The Ve oeraor Le A ( a ) ( A B )( C D ) ( A C ) ( B D ) L a be a ( ) marx The ve ( A ) s he veor obae by sag he olums of A amely a ve( A ) : a he ( ) veor If B s ( q) C s ( q s) a D s ( s ) he ve( A ) ve( B ) rae( A B ) ve( A B C ) ( C A ) ve( B ) rae( A B C D ) ve( D )( A C ) ve( B ) Dsrbuos of Tes Sass We erve asymo exasos for he srbuos of es sass meoe revous seo u o he orer whh a be realy obae from he srbuo of Hoellg s geeralze rero Suose ha x ( L ) sasfy he Cramer s oo a have he

8 YOSIHITO ARUYAA eghh mome Frs we shall efe he hr a fourh orer umulas of Cx Le A be a oformable marx of osa a le z ( z z ) be a raom veor wh E [] z a Cov[ ] I orer umulas are eoe as L z Whe he hr a fourh κ ab E[ z z z ] : a b E[ z z z z ] ( δ δ δ δ δ δ ) κ ab : a b ab a b a b where δ aa a δ ab f a b he measures of mulvarae sewess a uross a be exresse as ( z) : κab rae( E[ z ve( z z)] E[ ve( z z) z] ) a b ( z) : κaaκbb rae( E[ z ve( z z)] ve( I ) ve( I ) E[ ve( z z) z] ) a b aabb a b ( z ) : κ E[( zz) ] ( ) Noe ha ( z) a ( z) are he mulvarae sewess a uross resevely roue by ara [] a ( z) s aoher mulvarae sewess by Isoga [8] Usg he roeres of marx algebra Seo we have ( Az) rae( AAE[ z ve( z z)] ( AA ) ( AA )) E[ ve( z z) ]) z ( ) ( Az) rae AAE[ z ve( z z)] ve( AA ) ve( AA ) E[ ve( z z) ] z ( ) ( ( )) ( Az) E[( ( Az) Az) ] rae ( AA ) rae AA I s oe ha he es sass base o he rasforme oulaos / / Ax are affe vara Hee seg A ( C C) C obas he measures of mulvarae sewess a uross for he saarze Cx as follows

9 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS : rae( E[ z ve( z z)] ( ) E[ ve( z z) ]) (5) z : rae( E[ z ve( z z)] ve( ) ve( ) E[ ve( z z) ]) (6) z : E[( z z) ] ( )( ) (7) / / / / where : ( C )( C C) C If A ( ) he above measures a be smlfe as ( E[( / z) ]) ( ) E[( / z) ] ( ) Uer era mmal assumos lug Cramer s oo he exaso of he srbuo base o he Egeworh exaso s val as show by Chara a Ghosh [] Bhaaharya a Ghosh ([] []) e We summarze hree es sass efe Seo as T g The alyg (5) (6) a (7) o he asymo exaso of Hoellg s geeralze rero gve by Fujosh [5] he umulave srbuo fuo of T g a be exae as ( ) ( ) ( ) Pr T g y G y lg l y o l y ( ) ( ) ( ) y y G y G y o (8) ( )( ) where G ( y) meas he umulave srbuo fuo of a eral hsquare varable wh egrees of freeom a G ( y) s he ervave of G ( y) The oeffes l a he arameer are as follows: () If T g s T ( ) gve () he s ae o ( ) ( ) ( ) ( ) ( )

10 YOSIHITO ARUYAA ( )( ) ( ) ( ) ( ) a 8 8 () Else f g T s ( ) T gve () he s ae o The oeffes ( ) L l l are obae by subsug o he ase () () Else f g T s ( ) T gve () he s ae o a ( ) 6 ( ) ( ) Remar ha he srbuo fuo of ( ) T oes o ee o a ha of ( ) T s o relae o Whe hese exressos

11 ASYPTOTIC APPROXIATIONS FOR DISTRIBUTIONS 5 meoe above reue o he resuls obae by aruyama [5] We a also see ha he ase () s esseally he same as he asymo exaso for Hoellg s rero see Kao [] a Fujosh [] for more eals Geerally he Corsh Fsher exaso s use as a aroxmao o he rue ereage o Le τ α a u α eoe he rue ereage o a he ereage o of lmg srbuo of T g resevely ha s Pr( T g τ ) Pr( X u ) where α α X s a varae of a h-square srbuo wh egrees of freeom Alyg he geeral verse exaso formula by Hll a Davs [7] o (8) we a exa τ α as uα uα ( ) uα uα o ( )( ) I aual use we ee o relae he uow arameers a by her esmaors For esmaors of mulvarae sewess a uross see Kala [] ara [] a Isoga [8] Aowlegeme The auhor has he eor a referees for helful ommes Referees [] R N Bhaaharya a J K Ghosh O he valy of he formal Egeworh exaso A Sas 6 (978) -5 [] R N Bhaaharya a J K Ghosh O mome oos for val formal Egeworh exasos J ulvarae Aal 7 (988) [] T K Chara a J K Ghosh Val asymo exasos for he lelhoo rao sas a oher erurbe h-square varables Sahyà A (979) -7 [] Y Fujosh A asymo exaso for he srbuo of Hoellg s T - sas uer oormaly J ulvarae Aal 6 (997) 87-9 [5] Y Fujosh Asymo exasos for he srbuos of mulvarae bas sass a oe-way ANOVA ess uer oormaly J Sas Pla Iferee 8 () 6-8

12 6 YOSIHITO ARUYAA [6] S W Greehouse a S Gesser O mehos he aalyss of rofle aa Psyhomera (959) 95- [7] G W Hll a A W Davs Geeralze asymo exasos of Corsh-Fsher ye A ah Sas 9 (968) 6-7 [8] T Isoga O measures of mulvarae sewess a uross ah Jao 8 (98) 5-6 [9] K Io Asymo formulae for he srbuo of Hoellg s geeralze T sas A ah Sas 7 (956) 9-5 [] K Io Asymo formulae for he srbuo of Hoellg s geeralze T sas II A ah Sas (96) 8-5 [] T Iwasha Asymo ull a o-ull srbuo of Hoellg s T - sas uer he ellal srbuo J Sas Pla Iferee 6 (997) 85- [] Y Kao A asymo exaso of he srbuo of Hoellg s T - sas uer geeral srbuos Amer J ah aageme S 5 (995) 7- [] E L Kala Tesor oao a he samlg umulas of -sass Bomera 9 (95) 9- [] K V ara easures of mulvarae sewess a uross wh alaos Bomera 57 (97) 59-5 [5] Y aruyama Asymo exasos of he ull srbuos of some es sass for rofle aalyss geeral srbuos J Sas Pla Iferee 7 (7) [6] D F orrso ulvarae Sasal ehos (Fourh eo) Duxbury Press (5) [7] C R Rao Tess of sgfae mulvarae aalyss Bomera 5 (98) [8] G A F Seber A arx Haboo for Sasas Joh Wley & Sos New Yor (7) [9] Soa O he srbuos of he Hoellg s T - sass A Is Sas ah 8 (956) - [] Soa A asymo exaso of he o-ull srbuo of Hoellg s geeralze T - sas A ah Sas (97) [] S Srvasava Profle aalyss of several grous Comm Sas Theory ehos 6 (987) [] H Yaaghara Asymo exasos of he ull srbuos of hree es sas a oormal GANOVA moel Hroshma ah J () -6