Asymptotic Methods in Statistical Inference - Statistics Centre

163
• For the measure (18.1) we typically have ( ) →
Φ(), and this holds as well if ( ) is replaced by
a consistent estimate such as ( ˆ ). In such a
situation we could use Φ() as an estimate of
( ). This is certainly easier than estimating
( )by ( ˆ ) and approximating the latter
by ∗
. Which is more accurate?
• More generally, in our discussion of the bootstrap
we are taking limits over both and in order to
estimate ( ). But typically ( ) → ( ),
and then the theory of the last few lectures tells
us that this limit ( ) can be estimated by the
plug-in estimate ³ ˆ ´, with error
³
−12´.
In this case the error in estimating ( )is
0
= ³ ˆ ´
− ( )
= h ³ ˆ ´
− ( )
i
− [ ( ) − ( )]
=
³
−12´ − [ ( ) − ( )]