Claim The Kaplan Meierestimator is the NM LE por censoreddataIn order to show this we first write down the nonparametriclikelihoodfunction

L FILIP Ti ki LpCT ya ghi

Reran a remark we mentionedf these are ties of

censoringanddeaths then we always assumethat the censoring

happens beforedeath

Step One can showthatthe Nonparametric likelihoodfunex.im

is maximised by assigningmassprobability

LiMi to fail if I I

Yei Kiin if O

Step2 Using 47 we havethat I K's 49

L Cmi jimMi

gI mjThen we have e di faitmj

Let di SjMjThensj mj 4 dj es

andmi ii Ej mj di l dj 133

Plugging 121 and13 into 141 gives

I Eiichi's Hip fami Iii Iit ii Ipozyonprooni

gEismmraffinfB it FITit

Then we can findthat L is maximised atKj

I andtherefore mni.fi jEii njEiTn i 11which leads to that thecorresponding

811 1 Et I E Thjj Yjet

is the Kaplan Meterestimator

T The large sample propertiesof the KaplanMeher estimators

It a Set as n P consistency

SH is approximately distributed as asy normalityevent

NC sets s JotdFu aC Him5 Fct

pCT Etwwe FM I n

andI Heu 4 Fens C Gini

a1pcysul.tl 0CmirCTikIsn1pct u C n PC Tsn IPCC ng

6 The hazard funithnRecall that

Sct ee

where Ale is the cummulativehazard function

i e Alt got dads

Then a natural estimator Peterson of acts can bederived as follows

Htt agent ig fni 1the

i EEL

p.hr jEgee J h F The Nelson Aalenestimator

Note that logic x x X

7 Applications e gto estimate the mean life time

IP T m IP T E m then m is called the

mean life timeWe let 5451then we can solve the

atom ofnationand

calculate into estimate the probabilityof loving an

extra time

IP Tst Seti