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7.2 Estimation of Survival Function

(I) Parametric approach

Suppose are failure times corresponding to censor

indicators (

nttt ,,, 21 K

nwww ,,, 21 K ,1=iw death;

censoring). Then the likelihood function is

,0=iw

( ) ( )[ ] ( )[ ]( )

( )

( )

( )[ ] ( )

=

==

=

==

n

i

i

w

i

n

i

i

w

i

i

n

i

w

i

w

i

tSt

tS

tS

tftStfL

i

i

ii

1

11

1

(as

( )[ ] ( )[ ] ( )

( )[ ] ( )[ ] ( ) ( )iiw

i

w

ii

i

w

i

w

ii

tTPtStStfw

tftStfw

ii

ii

===

==

1

1

0

1

)

where ( ) ( )tSt , depends on some parameter . Then, the

parameter estimate can be obtained by solving 0

=

L.

Then, and can be obtained by evaluating( )t ( )tS at

.

Example:

Let T have exponential density. Then, ( ) ( ) ,, tt etSetf ==

and ( ) =t . Then,

1

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( ) =

=n

i

tw ii eL1

and further

( ) ( )[ ] ( )[ ]

( )

=

==

==

=

n

i

i

n

i

i

n

i

ii

tw

twLl

11

1

log

loglog

Thus,

( )

=

=

=

= ===

n

i

i

n

i

in

i

i

n

i

i

t

w

t

wl

1

1

1

1 0

.

(Intuitively, ( ) 1

=

TE the estimate for mean survival

time is

=

==n

i

i

n

i

i

w

t

1

1

1

) Then, ( )t

etS = . For example, in

the motivating example,

5.63

4

11671075.976

1111

1

1 =+++++++

+++==

=

=n

i

i

n

ii

t

w

.

Then, ( ) 5.634

t

etS

= .

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(II) Nonparametric approach

Let be death times. The number of

individuals who alive just before time , including those who are

about to die at this time, will be denoted , for

)()2()1( mttt

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since 1)1( = tTP .

Example (continue):

In the motivating example, we have

)( it 6 7 11

in 8 6 1

id 1 2 1

Thus,

( ) ( ) ( )

( ) ( ) ( )

=

=