Outline. Outline. Outline. Questions. Multivariate Normal Distribution. Multivariate Normal Distribution

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1 Multvrt orml Dstruto hyh-kg Jg Drtmt of Eltrl Egrg Grdut sttut of Commuto Grdut sttut of tworg d Multmd Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Outl ssssg th ssumto of ormlty Dttg Outlrs d Clg Dt Trsformtos to r ormlty Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Qustos Wht s th uvrt orml dstruto? Wht s th multvrt orml dstruto? Why to study multvrt orml dstruto? Multvrt orml Dstruto Grlzd from uvrt orml dsty s of my multvrt lyss thus Usful romto to tru oulto dstruto Ctrl lmt dstruto of my multvrt sttsts Mthmtl trtl 6

2 Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Qustos Wht s th formul for th rolty dsty futo of uvrt orml dstruto? Wht r th rolty mg of rmtrs d? How muh rolty r th trvls - + d - +? How to loo u th umultd uvrt orml rolty Tl d? Qustos Wht s th Mhlos dst for uvrt orml dstruto? Wht s th Mhlos dst for multvrt orml dstruto? Wht r th symol for d th formul of th rolty dsty of -dmsol multvrt orml dstruto? Qustos Wht r th ossl shs surf dgrm of vrt orml dsty? Wht s th ostt rolty dsty otour for -dmsol multvrt orml dstruto? Wht r th gvlus d gvtors of th vrs of? Rsult 4. Qustos Wht s th rgo tht th totl rolty sd uls -? Wht s th rolty dstruto for lr omto of rdom vrls wth th sm multvrtorml dstruto? Rsult 4. How to fd th mrgl dstruto of multvrt-orml dstruto y Rsult 4.? Qustos Wht s th rolty dstruto for rdom vtor otd y multlyg mtr to rdom vtor of rdom vrls wth th sm multvrt-orml dstruto? Rsult 4. Wht s th rolty dstruto of rdom vtor of multvrt orml dstruto lus ostt vtor? Rsult 4.

3 Qustos Gv th m d ovr mtr of multvrt rdom vtor d th rdom vtor s rttod how to fd th m d ovr mtr of th two rts of th rttod rdom vtor? Rsult 4.4 Qustos Wht r th f-d-oly-f odtos for two multvrt orml vtors d to ddt? Rsult 4.5 f two multvrt orml vtors d r ddt wht wll th rolty dstruto of th rdom vtor rttod to d? Rsult 4.5 Qustos rdom vtor s rttod to d th wht s th odtol rolty dstruto od gv =? Rsult 4.6 Wht s th rolty dstruto for th sur of th Mhlos dst for multvrt orml vtor? Rsult 4.7 Qustos How to fd th vlu of th Mhlos dst for multvrt orml vtor wh th rolty sd th orrsodg llsod s sfd? Rsult 4.7 Qustos Wht s th sh of h-sur dstruto urv? How to loo u th umultd hsur rolty from Tl d? Wht s th ot dstruto of two rdom vtors whh r two lr omtos of dffrt multvrt rdom vtors? Rsult 4.8 Uvrt orml Dstruto f [ ] 8

4 4 9 9 Tl d Tl d ur of Dst ur of Dst Mhlos Mhlos dst dst -dmsol orml Dsty dmsol orml Dsty ] [ sml from rdom vtor s f Eml 4. Eml 4. vrt vrt orml orml Corr r r E E Eml 4. urd Dst Eml 4. urd Dst ] [ 4 4 Eml 4. Dsty Futo Eml 4. Dsty Futo ]} [ { f

5 Eml 4. vrt Dstruto Eml 4. vrt Dstruto = = 5 = =.75 6 s Cotours Costt rolty dsty otour ll suh tht - surf of llsod trd t Rsult 4. ostv dft - for for ostv dft Eml 4. vrt Cotour Eml 4. Postv Corrlto vrt orml gvlus d gvtors [ [ ] ] 9 5

6 6 Prolty Rltd to Prolty Rltd to urd Dst urd Dst rolty hs vlus stsfyg old llsod of Prolty Rltd to Prolty Rltd to urd Dst urd Dst Rsult 4. Rsult 4. must for vry 4 4 Eml 4. Mrgl Dstruto Eml 4. Mrgl Dstruto dstruto of Mrgl ] [ ] [ 5 5 Rsult 4. Rsult 4. d d 6 6 Proof of Rsult 4. Prt Proof of Rsult 4. Prt vld for vry y lr omto

7 7 7 7 Proof of Rsult 4. Prt Proof of Rsult 4. Prt s rtrry d d d d d d 8 8 Eml 4.4 Lr Comtos Eml 4.4 Lr Comtos vrfd wth Y Y 9 9 Rsult 4.4 Rsult 4.4 Rsult 4. t Proof 4 4 Eml 4.5 ust Dstruto Eml 4.5 ust Dstruto Rsult 4.5 Rsult 4.5 ddt d oly f ddt f Cov ddt 4 4 Eml 4.6 dd Eml 4.6 dd d lso s ddt of r ddt d ot ddt 4

8 8 4 4 Rsult 4.6 Rsult 4.6 ovr d wth m orml s gv dstruto of odtol Proof of Rsult 4.6 Proof of Rsult 4.6 wth ovr ot orml Proof of Rsult 4.6 Proof of Rsult 4.6 gv ddt r ddt d f f P P P P Eml 4.7 Codtol Eml 4.7 Codtol vrt vrt show tht f Eml 4. Dsty Futo Eml 4. Dsty Futo ]} [ { f Eml 4.7 Eml 4.7 ] [ f f f

9 Rsult 4.7 Rsult 4.7 dstruto th rtl of th th ur dots whr }s { Th rolty sd th sold llsod 5 5 Dstruto Dstruto Gmm dstruto wth frdom d.f. dgrs of ; f Z 5 5 Dstruto Curvs Dstruto Curvs 5 5 Tl d Tl d 5 5 Proof of Rsult 4.7 Proof of Rsult 4.7 Z Z Z Z Proof of Rsult 4.7 Proof of Rsult 4.7 dstrutd y w rdom vrl ssgd to th llsod y th rolty s P P

10 55 55 Rsult 4.8 Rsult 4.8 wth ovr mtr ot orml r d d mutully ddt Proof of Rsult 4.8 Proof of Rsult 4.8 dgoltrmsof off trms of lo dgol ] [ Eml 4.8 Lr Comtos Eml 4.8 Lr Comtos 4 ddt dtl Eml 4.8 Lr Comtos Eml 4.8 Lr Comtos Cov 6 Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Qustos Wht r rdom smls? Wht s th llhood? How to stmt th m d vr of uvrt orml dstruto y th mmumllhood thu? ot stmts Wht s th multvrt orml llhood?

11 Qustos Wht s th tr of mtr? How to omut th udrt form usg th tr of th mtr? Rsult 4.9 How to rss th tr of mtr y ts gvlus? Rsult 4.9 Rsult 4. Qustos How to stmt th m d ovr mtr of multvrt orml vtor? Rsult 4. Wht s th vr rorty of th mmum llhood stmts? Wht s th sufft sttsts? Mmum-llhood Estmto Multvrt orml Llhood Jot dsty of s futo of llhood rdom sml from Mmum llhood stmto Mmum llhood stmts d for fd 6 64 Tr of Mtr tr ; tr tr tr tr tr tr tr tr tr d tr s slr 65 vtor tr Rsult 4.9 symtr mtr tr tr 66

12 67 67 Proof of Rsult 4.9 Proof of Rsult 4.9 tr tr tr tr tr tr tr tr mtr mtr C C C C C C m m m m m Proof of Rsult 4.9 Proof of Rsult 4.9 dg tr tr tr tr Λ ΛPP ΛP P Λ P P ΛP P Llhood Futo Llhood Futo tr tr L 7 7 Rsult 4. Rsult 4. holdg oly for wth ulty ostv dft for ll ostv slr symmtr ostv dft mtr tr 7 7 Proof of Rsult 4. Proof of Rsult 4. suh tht ttd wh ur oud s t mmum hs tr ostv ll gvlus of tr tr tr tr tr 7 7 Rsult 4. Mmum Llhood Rsult 4. Mmum Llhood Estmtors of Estmtors of d d ˆ ˆ rdom sml from

13 7 7 Proof of Rsult 4. Proof of Rsult 4. L L ˆ ˆ ˆ tr Eot of tr vr Prorty vr Prorty r MLE of ˆ ˆ MLE of ˆ ˆ ˆ MLE of Emls mmum llhood stmtor of ˆ mmum llhood stmtor of ˆ h h ufft ttsts ufft ttsts oulto orml multvrt sttsts of r sufft d d through osrvtos dds o th whol st of Jot dsty of tr Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Qustos Wht s th dstruto of sml m for multvrt orml smls? Wht s th dstruto of sml ovr mtr for multvrt orml smls? Dstruto of ml M Dstruto of ml M Rsult 4.8 f. Multvrt s Uvrt s rdom sml from

14 mlg Dstruto of rdom sml from Uvrt s s - s Multvrt s Z Z Z Z Z Wshrt dstruto W - 79 w Prorts W Wshrt Dstruto ostv dft m W m m W W CC W CC CC m 4 m tr m 8 Outl troduto Th Multvrt orml Dsty d ts Prorts mlg from Multvrt orml Dstruto d Mmum Llhood Estmto Th mlg Dstruto of d Lrg-ml hvor of d Qustos Wht s th uvrt trl lmt thorm? Wht s th lw of lrg umrs for th uvrt s d th multvrt s? Rsult 4. Wht s th multvrt trl lmt thorm? Rsult 4. Qustos Wht s th lmt dstruto for th sur of sttstl dst? Uvrt Ctrl Lmt Thorm dtrmd y lrg umr of uss rdom vrls hvg romtly th sm vrlty hs rly orml dstruto ddt s lso rly orml for lrg sml sz 84 4

15 Rsult 4. Lw of Lrg umrs Y Y. Y ddt osrvtos from oulto my ot orml wth E Y Y Y Y Y ovrgs rolty to Tht s for y rsrd P[ -ε Y- ε] s Rsult 4. Multvrt Css ddt osrvtos from oulto my ot multvrt orml wth m E ovrgs rolty to ovrgs rolty to Rsult 4. Ctrl Lmt Thorm ddt osrvto from oulto wth m d ft ovr s romtly for lrg sml sz ut good romto for modrt wh th rt oulto s rly orml 87 for lrg - s lrg for lrg - Lmt Dstruto of ttstl Dst rly for lrg sml sz romtly los to wth hgh rolty wh romtly 88 Outl ssssg th ssumto of ormlty Dttg Outlrs d Clg Dt Trsformtos to r ormlty Qustos How to dtrm f th smls follow orml dstruto? Wht s th Q-Q lot? Why s t vld? How to msur th strghtss Q-Q lot? 5

16 Qustos How to us Rsult 4.7 to h f th smls r t from multvrt orml oulto? Wht s th h-sur lot? How to us t? Lt lr s orml dstruto Q-Q Plot osrvtos o dstt d modrt to lrg.g. Porto of z P[ Z ] dz Plot to s f thy r romtly f th dt r from 9 Eml 4.9 Eml Hstogrm of MdTrm ors of tudts of Ths Cours 6 Q-Q Q Plot of MdTrm ors of tudts of Ths Cours 6 95 = r Q =

17 Eml 4. Rdto Dt of Closd-Door Mrowv Ov Msurmt of trghtss r Q Rt th ormlty hyothss t lvl of sgf f r vlu Tl 4. Q flls low th rort Tl 4. Q-Q Plot Corrlto Cofft Tst Eml 4. For dt from Eml Q Q r r.95 Do ot rt ormlty hyothss 99 Evlutg vrt ormlty Eml 4. Ch f roughly 5% of sml osrvtos l th lls gv y ll suh tht.5 7

18 [ ] d Eml d 97.9 v out of osrvtos r wth Grtr th 5% rt vrt ormlty Howvr sml sz s too smll to rh th oluso 5 d.9 5 d ot tht Ch-ur Plot Ordr th surd dst d utl of th h -sur dstruto wth dgrs of frdom Grh ll d Th lot should rsml strght l through th org hvg slo surd dst d d 4 Eml 4. Ch-ur Plot for Eml 4. Eml 4. Ch-ur Plot for Eml Ch-ur Plot for Comutr Grtd 4-vrt 4 orml Dt Outl ssssg th ssumto of ormlty Dttg Outlrs d Clg Dt Trsformtos to r ormlty 7 8

19 ts for Dttg Outlrs M dot lot for h vrl M sttr lot for h r of vrls Clult th stdrdzd vlus. Em thm for lrg or smll vlus Clultd th surd sttstl dst. Em for uusully lrg vlus. h-sur lot ths would ots frthst from th org. 9 Outl ssssg th ssumto of ormlty Dttg Outlrs d Clg Dt Trsformtos to r ormlty Qustos How to trsform sml outs roorto d orrlto suh tht th w vrl s mor r to uvrt orml dstruto? Wht s o d Co s uvrt trsformto? How to td o d Co s trsformto to th multvrt s? Qustos How to dl wth dt ludg lrg gtv vlus? Hlful Trsformto to r ormlty Orgl l Couts y Proortos ˆ Corrltos r Trsformd l y ˆ logt ˆ log ˆ r Fshrs z r log r o d Co s Uvrt Trsformtos l Choos λ to mmz l l 4 9

20 Eml 4.6 vs. Eml 4.6 Q-Q Plot 5 6 Trsformg Multvrt Osrvtos owr trsformtos for l λ ˆ th hrtrsts lt to mmz ˆ ˆ ˆ ˆ ˆ ˆ l 7 Mor Elort roh owr trsformtos for th hrtrsts lt λ to mmz l λ λ s omutd from λ l 8 Eml 4.7 Orgl Q-Q Plot for O-Door Dt Eml 4.7 Q-Q Plot of Trsformd O-Door Dt 9

21 Eml 4.7 Cotour Plot of for oth Rdto Dt Trsform for Dt ludg Lrg gtv lus log log