2 Estimation

fz min Ti Ci

Data Hi f i i in ind

where Yi T f If Tiecilgift

s.ee i FIzfAssumption Ti iidn F Cf Sland Ti I Ci 11 this assumption canbe relaxed

11 Tiff where X is arecovariates

Asfor the obesservation i we hone

Li fly it if fi I

Sigi Pti Yi if fiThen we have

II Hey g tiffeyDti

RI In fart denote Ci I't G CfThen the full likelihood is

EE II ffyiif Gcy.nl Jti FcyijgcyrThis is to say themultiplication is

done to

µ the independence of Ti andCiEY fly JC Glyn if f e T se

Three

I it o OtisRI if fi I that means Yi Ti

then Li fyi n

Ah Y Exp X Yi i I in

Li I facycont

Yu Poisson Al dhereter iffiao

Liat lPxcY Yi Ti Yi1pcti TiScifiFcyit

Rh dont'd I

It II ish Gy i Tryin gig GJ't

iiNow if ft has paramerer

0 them

Fine is definedas

arygmaxiITLfiyijJti Fiy.n yti

Mfmlost't T't't T

Then the rest is tastesame as the standard Arlit results

for instancern E Is Arco Ico's

Ex Exponential Ti iidExped

Note that Mi

Li fly it de if di L

i Fyi Say eMi if ti o

Then the likelihood

IT Li Eddie III e

i'Isif e

de y j

Ine anqm.geL qq.mg

kiIdi hyddkiayiI4

ThenFarce I Ein

Einylet's introduce the notation nu GI fiConfidence intervals

If nu n i.e no censored data

then I Isit Yi

S.me Ti i'd ZxpedH Sii Ti n Han

Thensign Xian

CI 7 I is aanf.int hamae

1 Where there are censoreddata

It follows from the CLT or the asymptotic normality

of MLF that

F is asymptotically distributed as

fN d 1

since 24mg LF YI

variant f that ITg

rThen

io na c E2 l Delta method

Note that the asymptotic variane ofF

is which is afunctionof the unknowparameter d

Variance stablisationAssume Y is a r v with mean ph and vanname

T Assume fl j is a continuous funition satisfying

regularity conditions

ThenglY has mean gyu and vanname

yn J oh

This is due to the Taylor expansionthat

gly Fu gyu t g'yn LY1hThen Ecgets rig Cfn l

Van gill Cg'yrijvarithff'qmJ r

5 00

Then we choose the function fix hf'dThen due to the Delta method

Ellvge Is lugedvaruge.in ehIn ti InL

2.2 Estimation of Sct

Sime Siti exp f ithit follows from the Delta method

Ect empt It gcxI exp txt

Et 8h17 exp L de due to 41

Mdvomfiti

yinCerpetticityNa E expc sit1