Outline1Testing with nuisance parameters

2 UMPU multivariate tests

3 Conditioning on null sufficient stat

Nuisance Parameters

Commonset Extra unknown parameters

which are not of direct interest

F Po O a Er Ho O us H O

O parameterof interest

X nuisance

issueX unknown but might affect

type I error or power of a given test

EI X xniiidNCn.ci Y ym Nlu E

M V 02 unknown

Ho M V vs H in U

O u u to or or n E

EI X BinonCn t Xz Binomcnz.ithi Nz known not nuisance parametersHo IT E Tz Vs H T Tz

MultiparameterExpfamilies

Assume X n

fo C EO'T X 4G Aco Dh

DE IRS XelRr both unknown

How to test Ho OE o vs H c OH

idea Condition on UCX to eliminate dep on X

1 Sufficiencyreduction

TH UCH ngo tin

EO't In Aceg t n

density wrt ut n e.g Lebesgue on Rst2 Condition on UCX

go CtuCq ttugqo.iz.us de

ee't t.tn x

gct.in

eBd03 fe tt ADgCz u dz

O't Buttg

t ue

3 CondiestTest Ho OE us H OE in

S parameter model Q fqoCtlD OEu

Nyote if s l this family has MLR in T

Even if s I we still have gotten rid of X

theorem Let P be full rank exp fan with

densitiespg eOTCD th UG ACO

hc

OE IR XE Rr CO 5 ER open Oo possible

a To test Ho OEOo us It O Oo there is a UMPU

test 0 4 y TG UG where

Htin In IO t e cCu

With da gcu chosen to make

Eq 0 671467 1 0

b To test Ho O Oo us Hi O too there is a UMPU

test of x YET i 467 where

yct.ruI tacku or t cdnZiCu t CienO E E Cc Cu Cdn

with d u go.cn chosen to make

IEq 04 7 lucid if

Foo 0 4 o UK if O

Note h has disappeared from the problem

Ex Xi id Poison 5 1,2

Ho M E Ma us H M Me

pix IImÉÉexe Ya let te

y

Where yo log Mi Ho Y E z H z

ei iÉxii an

Ho OEO us H O I

Reject for conditionally large values of

X given X Xz u

Polxix un ett

MIn xil EcX O

9 ex xTÉx

Biron u oe Yu

Bison a YySo in the

end we do a

Binomial test

ProofsketdyX n r

i

70OO

1 Any unbiased test has PCO d L htt

continuity

2 Power D on boundary Eq 0 IU o

UK complete sufficient on boundary submodel

3 Of optimal among all tests with conditional level 0

by reduction to univariate model

Proof Assume of any unbiased test

1 Eg HI E l c a f DerKeener Than2 4

Eo 04 infinitely diff on R can diff under f0 unbiased Eq IOCx5f V CQ d er

Stef Boundary subm.de Pq fPq Oo HER

pg Cx eUk Ace

qYPq is full rank s param exp fam UCX come.su

Let flu Eo.coCx3lUCXI a d

Ee CHUM Eo Cx r O th

flu O

Eg.coCx31UCxs a O tu

dTwo.si e gCu qfEq 0lU u

EqflT EooCtiu3 o lU

Eo T ol a In

Eq glu Eo TG a fz BoCoo _Of

oEoo Olu o Konda rotas

Steps Forany

value u the conditional model is

tinOt Belo

glt.ir 1 param exp fan

In one two sided case we have shown

y tin is UMP UMPU in Qu

Let g tin E 0cxjltcxt t.ua LEo Ift n IU if Eo 0Cx lucx u

y if 0 00

OTC D is a fond.ie test of Ho us H

in Qu with powerL at boundary

One sided case Cor 0 00

4ft u is the UMP test of 0 0 us 0 0

in Qu which is a l param exp fan

Two sidey Ct D

is the UMP test of 0 9 us too

amongtests with power D over _0 Oo

Keener Than 12.22 main thm for two sidedtests

In either case y has higher coed powerthan a s

For O X E R i

Eo x Eo Eo ECT n l u

E Eo Eo yet n l u

Eo 0 41

EI X Xn El Ncn E 030 unknown

Ho u O us Hi M 0

II iris

for odx EZEK E EXE IE g

Condition on U X 11 112

If µ O f is rotationally symmetric

11 11 117 u Unit sin 5

Unit Sm indef of 11 11

Optimal test rejects when EXi is extreme given 11 11

Let 5 I X x

Ex 25 Exit NID

11 11 nFE i In'x

n 1 5 11 112 1 511Prof X 112

In

Xz d2

Xp Xz

NNG.no

ITN

innext lecture

x

res when

is extreme given 11 11

under Horej when If is extreme

htt next topic

PermutationTestsEven if we don't get a UMPU test at the

end conditioning on null suff stat still helps

EI Xi gl nniidPY YmiidQHo P QHi PtQ

Under Ho F Q X Xn t Ym P

Let 2mm X ok Y Yn

Under Ho U Z Za Zenon comet su

Let Sam Permutations on ntm elements

X y I U t Unit E U i te Sn m3

Thus for test stat T if P Q

Pp TC at Iu my ESfICtZ zt3

Monte Carlo test In practice we sample

IT ITBd

me g B 1000

Then Z it t it t id Unif mU underHo

MC p value f Fe 1 ITCZ ETCitsz

Unit fits 13 if noties

p Z UnifC if there are ties