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Statistical Methods with Applications to Demography and Life Insurance
book

Statistical Methods with Applications to Demography and Life Insurance

by Estate V. Khmaladze
March 2013
Intermediate to advanced content levelIntermediate to advanced
242 pages
5h 47m
English
Chapman and Hall/CRC
Content preview from Statistical Methods with Applications to Demography and Life Insurance
112 Kaplan–Meier estimator for F
so that I
{|δm
n
(t)|
2
>ε}
= 0 for all t, as soon as 1/n < ε, and the sum
becomes equal to 0. To establish (b), note that
D
1
n
M
n
E
(t) =
1
n
hM
n
i(t) =
Z
t
0
Y
n
(s)
n
µ(s)ds. (10.6)
But
Y
n
(s)
n
=
1
n
n
i=1
I
{
e
T
i
s}
is just the tail of the empirical distribution function based on indepen-
dent and identically distributed random variables
e
T
i
. Therefore from
the Glivenko–Cantelli theorem (see Lecture 3), it follows that
Y
n
(s)
n
[1 F(s)][1 G(s)] n
uniformly in s with probability 1. Therefore, for all t > 0,
D
1
n
M
n
E
(t)
Z
t
0
[1 F(s)][1 G(s)] µ(s) ds
=
Z
t
0
[1 G(s)]dF(s) = A(t)
with probability 1, and not only in probability. Hence the limit theorem ...
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Publisher Resources

ISBN: 9781466505742