Asymptotic Statistical Analysis on Special Manifolds (Stiefel and Grassmann manifolds) Yasuko Chikuse Faculty of Engineering Kagawa University Japan

Transcription

1 Asyptotc Sttstcl Alyss o Specl Mfols Stefel Grss fols Ysuo Chuse Fculty of Egeerg Kgw Uversty Jp

2 [ I ] Stefel fol V ef { -fres R ; -fre set of orthoorl vectors R } ; ' I }. { Deso of V +.

3 Ex: O V : orthogol group. V : ut hypersphere R. crcle 3 sphere Applctos Erth Sceces Mece Astrooy Meteorology Bology. : A lrge lterture exsts for lyss of rectol sttstcs. 3 : vector crogr orbts of coets. 3

4 [ II ] Grss fol G ef { -ples R.e. -esol hyperples cotg the org R } G To ech -ple correspos uque orthogol projecto trx P P epotet of r. G P We crry out sttstcl lyss o P. 4

5 Ex: G {xes or urecte les through } Applctos: the sgl processg of rr wth eleets observg trgets. A rther ew sple spce. We cofe our scusso ly o the followg. V 5

6 [ III ] Populto strbutos o V. Ufor strbuto orlze vrt esure [ ] vrt uer H H' H O H O. 6

7 [ ] / V where the fferetl for j j < j choosg + x x x + x O x' + j x x ' x for x x V x x x s.t. π / / Γ / wth Γ S> etr S S + / S π / 4 ΠΓ /. ultvrte G fucto 7

8 . Mtrx Lgev L ; F strbuto pf f x etr F' / F / ; F' F / 4 for F trx w.r.t. [ ]. F ~ N M ; I Σ wth F MΣ ' I. : hypergeoetrc fucto wth trx rguet. Expoetl type strbuto. Pretrzto of F ΓΛΘ' sv where ~ Γ V Θ O Λ g λ λ wth frst row postve λ λ. 8

9 Moe: ΓΘ ' V Cocetrtos: Λ F etr F ' etr F ' etr ΛΓ ' etrλ. Θ Assue r uless otherwse stte. F Λ or : ufor strbuto. Hstorcl bcgrou: : rectol srtbuto 3 Fsher strbuto 953 Lgev 95 vo Mses strbuto 98 geerlzto by Wtso & Wlls

10 Mtrx-vrte orl M ; Σ Σ strbuto Z Y N f hs the pf.e. ll eleets of re... Let [.e. ] for ϕ the ~ Σ / N Z Z Σ ; I / π + M I / Z Z exp[ tr Z' Z Σ / ] N. / M Σ > Σ Y N M ; Σ. Z ~ Σ Y M > Σ /

11 Hypergeoetrc fuctos wth syetrc trx rguet. Strtg wth for syetrc Lplce trsfors S etr S F. ; ; ; ; / Y b b F S YS b b F S S etr q p q p S q p q p + > + Γ! ; ; l S C b b S b b F q p l l q p q p λ λ λ λ λ λ ト Σ Σ S

12 where orere prtto of... l l λ l S C l l l l l l l λ λ + + Π Σ wth : zol polyol wth syetrc trx S.

13 3. Mtrx Bgh strbuto pf f x etr B / F / ; / ; B B for syetrc trx. ~ N ; Σ I wth B Σ ' Expoetl type strbuto. Pretrzto of B ΓΛΓ' s where ~ O Λ g λ λ λ λ Mult-oe for wth Cocetrtos: B ; B ' Γ. Γ H Λ. Γ y I 3. Γ Γ Γ H O

14 4. A geerl for of strbutos pf f x p+ F p F q+ q B : p p ; / ; b b b b q ; ' B q / ; for syetrc trx. B Ex: F ' B etr ' B F b; ' B I ' B b. 4

15 5. Dstrbutos wth pf of the for f P ν where subspce of of eso ν : Pν Ex: L ; F ΓΛΘ' wth : etr B P ν ΓΓ' ν : R orthogol projecto trx oto wth pf etr F' ΘΛΓ' ΓΓ' etr F ' P q ν. ; B ΓΛΓ ' wth pf etr ' B etr ' ΓΓ' ΓΛΓ ' ΓΓ' etr[ P ' B P ] subspce spe by the the colus of Γ. ν ν ν 5

16 6. A etho to geerte ew fly of strbutos For ro trx Z Z / / / Z' Z Z' Z H T Z Z Z H / Z TZ polr ecoposto of Z. : oretto of Z. H Z V Wht s the strbuto of? / π usg Z H Z Γ / where Z [ H ]: Z T H Z [ ] T Z Z : Lebesgue esures orlze vrt esure o V. 6

17 Ex: Whe ; Σ I Z ~ N the f H Z / ' / H Σ H Σ H : Z More geerlly whe for the Z Mtrx gulr cetrl Guss [MACG Σ] strbuto. MACG ufor strbuto f Z I Z / / ' Z Σ g Z Σ Z Σ g H ' Z' Σ ZH H O H Z ~ MACG Σ. 7

18 [ IV ] Iferece for F ΓΛΘ' of Lgev strbuto Let ro sple fro L ; F. l where H sv wth H ' H O g x x M.l.e. s Σ where / the the log-lelhoo s [ ] F; tr F ' log F / ; Λ / 4 [ ' log / ; / 4 ] ' tr H ΘΛΓ H F Λ ~ H V x > > x. > Γ ˆ H log ˆ ˆ ˆ λ Θ H Λ g λ F ˆ ˆ / ; Λ / 4 x. * ˆ λ 8

19 [ V ] Asyptotc theores Iferece srtbuto theory for lrge sple lrge or sll cocetrtos er uforty hgh eso Λ 9

20 [ V. ] Lrge sple syptotcs Testg Z H H : Λ : F N ~ : N x x; uforty gst / F ; I I trz ' Z : ~ or Λ / Λ / F ; I I uer uer H H ; Λ H / tr uer for lrge H Uer for lrge. syptotclly Rylegh-style lelhoo rto Ro score loclly best vrt tests.

21 [ V. ] Sll or lrge Λ syptotcs Solve * by usg the syptotc expso for the ˆ F / ; Λ / 4 wth Λˆ sll or lrge: ˆ λ x + O Λˆ 3 Λˆ for sll. ˆ Σ + O Λ x ˆ λ j j ˆ ˆ λ + λ j for lrge ˆΛ. whe Λ hs r the λ + O Λ ˆ x ˆ

22 [ V.3 ] Hgh esol syptotcs Bcgrou Applctos copostol t cert perutto strbutos. For Lgev strbutos wth t s ow tht prctclly cocetrtos for 3 re lrger th those for. L ; F ~ : ufor strbuto F β F O? s see lter.

23 Asyptotc expsos for strbutos For ro sple fro L ; F put Z Ψ' q for fxe Ψ q Vq q fxe. pf f Z q Z ϕ Z { + tr F' ΨZ + [ 4 λト tr λ H q Z λ + tr Ψ' FF' Ψ] + O 3/ F' ΨZ } 3

24 where q q / ϕ Z π etr Z' Z / H H q λ : Herte polys. ssocte wth the orl N ; I I efe by q q q q Z Z C Z Z q ϕ ' ϕ Z λ Z wth Z Z Z. Z : ~ Nq j L ; F ~ for λ j ; I I q ot epeet o F : ufor strbuto s. 4

25 Geerlze St s Lt Theores Hstrcl bcgrou e Fett s theore 99: fte exchgeble sequece of ro vrbles s xe... Pocre s theore 9 or St s frst theore 98: syptotc orlty of the frst coortes of ro pot ufor o V s. If s orthogolly vrt hs -xture strbuto of orl σ N σ s [Dcos Eto Free Lurtze 98 s]. 5

26 St s frst theore Theore. [St 98] Asyptotc orlty of fte uber of coortes of the ufor vrte o V s. Theore. [Wtso 983] If ufor o V fte uber of eleets of re... N s. ~ 6

27 Extesos to ore geerl strbutos f P wth pf ν where : the orthogol projecto trx oto Ex: P ν ν q. subspce of eso L ; F ΓΛΘ' sv wth pf etr F' P ν B ; B ΓΛΓ' s wth pf etr P ' B P wth P ν [ ] ν ΓΓ'. ν 7

28 Theore. [Chuse 99] If ~ f P ν wth P ν ΓΓ' Γ V q fxe q the Γ q ~ N ; I I s. q ' : q q ufor cse 8

29 Theore 3. [Chuse] Suppose tht Whe β P. ~ f ν β < Γ' ~ : N ; q Iq I s. ex: β < Theore. β f. Whe the ltg strbuto epes o Ex: Whe ~ L ; F F ΓΛΘ ' the N Γ ' F ; I I Γ ' ~ : q q s. Whe ~ B ; B B ΓΛΓ ' the Γ' ~ : N ; I Λ I s. q q 9

30 3 St s seco theore lt orthogolty of Theore. [St 98] O V. Theore. [Wtso 983] If s ro sple fro the ufor strbuto o V the ' j < j epeet orl re utully N ; I I s. 3

31 Extesos to o-ufor strbutos o V Theore. [Chuse 993] If s ro F sple fro or ' the j < j re utully epeet orl N ; I I s. L ; B ; B 3

32 Theore 3. [Chuse] If s r. sple fro L ; β F β where Σ or ; β ' B B β < the j < j re utully epeet orl N ; I I s. If s r. sple fro B ; β B β the ' / Σ j < j re utully epeet orl N ; I I s B >. I < ex: β Theore. 3

33 Applctos Let be ro sple fro or ' The : N ; I I epeet s. L ; β F β B ; β B β <. j ~ 33

34 S where Ex: stce betwee ro pots Y + + : ~ tr Y ' Y N S ' ' j < j ' uj tr j ~ : N. S S ~ : N + s trs < j u S j +. + O s. tr 34

35 35 ~. ~ ' ' q q j j j trm q tr χ χ < ' ' ' < + j j j tr trm M. s : : where

36 Refereces [] C. Bgh A tpolly syetrc strbuto o the sphere A. Sttst [] Y. Chuse Sttstcs o Specl Mfols Lecture Notes Sttstcs Vol. 74 Sprger New Yor 3. [3] P. Dcos D. Free A oze e Fett-style results serch of theory A. Ist Her Pocre Probbltes et Sttstque

37 [4] T. D. Dows Oretto sttstcs Boetr [5] A. T. Jes Norl ultvrte lyss the orthogol group A. Mth. Sttst [6] P. E. Jupp K. V. Mr Mxu lelhoo esttors for the trx vo Mses-Fsher Bgh strbutos A. Sttst [7] C. G. Khtr K. V. Mr The vo Mses-Fsher trx strbuto oretto sttstcs J. Roy. Sttst. Soc. B

38 [8] R. J. Murhe Aspects of Multvrte Sttstcl Theory Wley New Yor 98. [9] M. J. Pretce Atpolly syetrc strbutos for oretto sttstcs J. Sttst. Pl. Iferece [] A. J. St Lt theores for ufor strbutos o spheres hgh esol Eucle spces J. Appl. Prob [] G. S. Wtso Lt theores o hgh esol spheres Stefel fols : S. Krl T. Aey L. A. Goo Es. Stues Ecooetrcs Te Seres Multvrte Sttstcs Acec Press New Yor 983 pp