Outline1 Hypothesis testing2 Ney m n Pearson Lemma

HypothesistestingIn hypothesistesting we use data X to infer

which of two submodels generated

Model D Po Oe

Null hypothesis Ho 0 E o

Alternative hyp H O c

henever H unspecified assure too

Ho is default choice we either

1 accept Ho fail to reject no definite cord

2 reject Ho conclude false true

EI X NCO 1 Ho 0 EO vs H O o

or Ho 0 0 vs H 0 10

EI Xi Xn P Y Ym Q Ho P Q vs Hi Pto

Common conceptual objection we know O O or

P 4 Q already why botherWe will return to this

Power Function

Can describe a test formally by its

critical on a.k.a testfunction

0 4 JO accept HoITE 0,1 reject v p IT1 reject Ho

In practice randomization rarely used X 10,13In theory simplifies discussions

For non randomized 0 the rejectionregion is

R x 41 7 13A XLR called acceptnee region

Powerfunction 1310 Eo OCK rejection probIpo Reject Ho

fully summarizes behavior of testThe significancelevely of is fenfo.to 3ofO

We say is a level a test if itssignificance level is E d C 0 I sorry

Ubiquitous choice is 9 0.05

Most influential offhand remark in history of science

Goal Maximize Pollo on subject to level a constraint

Examelesy

EI X NCO 1 Ho 0 0 H O to

Let Zo OI Ct d OI _normal calf

01 1 111 1 z 2 sided test

4 x 11 x to 1 sided test

x 11 X Zg or X Zay

a PCOI

i

o

Sometimes a unique best test exists

EI X NIO D Ho O so Hi O O

is best possible level a test

Bokoi

any othert

c o70

LikeliestA simple hypothesis is a single distribution

o Oo or 10When null alt both simple there exists a

unique best test which rejects fo largevalues of the likelihood ratio

06 3

11I

ypocx

C

O Efo x C

where f fo are nulllalt densitiesnote dominating measure always exists e.g Btp

and c j chosen to make ME CX a

is called the likelihoodrahotest LIT

Intuition discrete case

BANGPower under It fr f x dnCx __

Sig level Sp pix dnCx BuickAnalogy 100 to buy as much flour as possible

Neymas pea.rs

Proposition Keener 12 1

Suppose CIO ft maximizes Lagrangian

E 01 7 E 0 47

among all critical functions

If E 0 4 7 4 then 0 maximizes

E 0h17 among all level o tests

Essentially Lagrange multipliers work even thoughX infinite lots fete

Proof Suppose Eo 0CX so ThenE LOCH e E.co cxBtc6 EoC0cxB

e E.CO cxB cE.L0 cxi or

E 04 1

theorem Neyman Pearson LemmaLRT with significance level is optimal for

testing Hoi X po us H X p

Proof Maximize the Lagrangian

E.toCxD cEoL0cxD

fxlpCxs cpoCxDoCx dnCx

p

apololdu ftp.epolodnf To

Max first term X I when f x cpocx

Min second term X 0 when g x cpocx

Choose minimum C 5 t

Rolle a e de Pol za

And choose j to top up sig level

Poff c t pPoLf c Heeb PneDX

Keener gives converse up to wiggle room if

c for more than 1 value of X

Choosing q 2 for

Poffo d F

K like_caa

y u

d ii ii

p yCi Cz 3 3 ok too

Kelo 1 73 0

Keener Cor 12.4 n If fats f and is

LRT with level c 6,1 then IE 4G a

Proof u Sp pot nclp.iq both 70

IzI I'Fool la eoioien e.la

Ien

E E

O

C

E I 9 1 E to ftp.tolcl o HdnI Power I o f I

o

tx I panan exp fan

X X pixzTCx A z

e hCx

Ho z yo us It 2 2 Zo

itIeuII frocxi

y ETCxi n ACze

e7o ETC Xin A yo

eh Etcxi n ACz AG

X rejects for large ETCxo

an f I ETCX c

choose c z to make

ETCH c E z IPzo ETCxi c L

Surprise CX depends only on Z and

Sgn Z yo not on Zi Next topic