OutlineDX E and F distributions

2 Canonical linear model

3 General linear model

ReviewMultiparameter exp fans

OTCx X'UG ACOX e hCx

For Ho OEO vs H O Oo l side

or Ho 0 00 us H OF Oo 2 sided

The UMPu test 0 4 conditions on UCD

rejects for conditionally largelextreme TEX

Gaussian adjacent distributions

If Z Zd dN 0,1 thenshape scale

EZE XI Gamma 42,2

EV d Vad 2nd

As d as

I 51 PCIE H e 0 too

a Nld 2d

If Zn Nco D and V X'd ZIV then

retd Nco D as d 70

If V XI and K XIs V Vz then

V IdFd dz d XI as de a

Note if T td then TZ F.ae

Recall 2 nNd u E A c IRK'd.beRkAZ to NL Auto AEA

EI 1 sayle t test

X X El Ncu o7 nerd o o

We showed UMPU test of Ho M O us H M 0

rejects for large GI wheregz

I Lexi Ncm E5 1

E x ITn l

02L HX II RN XIrII IT E t

r is n 1

Why Orthogonal Projection 4 2

Xs N X

µII 11 11 41 711 ok

11km111 orarok Br X In Proj X

p

mm n1nn

7 X

s

I n I

Let Q g n Qr

where q In InEz En complete orthonormal basis

e.g via Gram Schmidt

X n Nn n In o In

New basis 1

Q'x n a IIIHQ'rXH HQxp Heixll

11 112 Nye Q'Q Inn 1 5

Q'X n N o Inh o

Zr Qi X N O o In

5 HZ.IT E X'mand 5215 we already knew from Basu

I

CanonicallinearModelf

a.me

fI.ynn l ozIn

dr n do d

Mo E IRI M E IRd

o 0

Test Ho u 0 us Hi M 0

or possibly one sided if d L

F xp Fam

plz eZ't E Zo Io 112112

1

Cord on Zo reject for largelsmalllextreme Z

Zo HZ test stat is Z Nlm o

Ei Io N o l Z testo

id reject for large 112,11117,114oz XI test

OZ unknown d I

Cord on Zo 112112 112,112 1112 112 1 241

Reject for largefsmalllextreme Z

Reject for large 71 11211

ZReject for largeqzI

E ta

Ct te st

id Reject for conditionally large HZ.lt

Reject for large 117114dL He f

FYd d dr

Here n XIfunctioning as estimator of o

IEE E Var 2

20ydCompare Z ZYo t Zyg

XZ 112,11 F 1172

LinealMany problems can be put into canonical

linear model after change of basis

Basic setup i

Observe Y NIO TIN or o

known or unknownTest O E

vs OE Iwhere

o E are subspaces of IRdimioto do dim D dotd

idea rotate into canonical formdo d n d

Q n Q Q Qrorthonormal o b for ab forbasis for n ot IR n

2 Q'Y Nn 8 8 o In

Ho Qi 0 0

Do Z X t or F test as appropriate

EI Linear Regression Xi C IRD fixed

Yi X B t ee e Nco E

Y Nd XP TIN X c IRD

Gin

Assume X has full column rank

O XB E Span Xi Xd

Ho 13 Bd _0 1 Ed Ed

O C Span Xd Xd

or 03 if d D

117112 HY Proj Y 112

HY xpno.si 2Bo.s argminHY XpH2

ECYi xipjzx'x x'y

Residual sunofsquares Rss

17,1121117112 HY Rojo.CH2

RSSo null Rss

F statistic is Zido Rss Rsscd do

HZrlf Cn d RSS kn d

n d called residualdegreesof freedom

d L Let Xo X XD E IRdo

Let X X Pro's Xi

X X x xD xo X

B Xie 4 11 11112 s e B 041 11

why O xp O Xup t Xp same p

E X'thx tl Q Y

t statistic E gB µRSSkn.FI Ie B

EI Two sample t test equal variance

Y Y id Ncn E y y dNCu ogI m until I n 1M

Model O EY no OE Sean CIN 1

Ho M _u c Span Imm

do L D 2 d ntm 2

orthogonalize HI

Reject for large

t.E.fi I.EYi I Ja

Eth aE FI

EI One way ANOVA

Ya Mk t Ek Ei dNco E

k I m i I n

Ho M Mm n

YI I EYa SI E Yai Tha

I 1 EEK SI I Y.si ItMh k i

do I D m d m Cn l

Rss Yai YI 114112 n ETIk

RSE Ye i F114112 min'T

RSE Rss n YI MI

n YI FI

F stat ME E y Ibetween variance

I ELY II within varianceinCn l