
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 ...