TBS903 Autumn2009 Subject Outline Subject Outline (Final Version)
Outline - stat.berkeley.edu
Transcript of Outline - stat.berkeley.edu
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
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