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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
Testing exponentiality 77
It is obvious that K
n
depends on λ . However, we have now remark-
able flexibility with substituting any consistent estimator
ˆ
λ, not nec-
essarily the maximum likelihood estimator, and not even necessarily a
n-consistent estimator:
suppose
n(
ˆ
λ
n
λ )
2
is small in probability:
n(
ˆ
λ
n
λ )
2
= o
P
(1), as n ;
then if we replace V
n
in (7.6) by
ˆ
V
n
(t) =
n[
b
F
n
(t) F
ˆ
λ
n
(t)],
the change in W
n
will be asymptotically negligible.
Indeed, consider a Taylor expansion in λ up to the second term:
n
F
ˆ
λ
n
(t) F
λ
(t)
=
λ
F
λ
(t)
n
ˆ
λ
n
λ ) + R
n
(t)
n
ˆ
λ
n
λ )
2
2
,
where the reminder term can be written as
R
n
(t) =
2
λ
2
F
˜
λ
(t) = t
2
e
˜
λt
and
˜
λ is a point between ...
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Publisher Resources

ISBN: 9781466505742