Asymptotic Methods in Statistical Inference - Statistics Centre

162
• Extend (i) to
⎛√ ⎞
³ˆ − ( )´
( )= ⎝ ≤ ⎠
( )
(18.1)
for some scale functional ( ). Now ( )isestimated
by ( ˆ ) and the latter is approximated
by ∗
. The total error is
where
∗ − ( )= ³ ∗ − ( ˆ )´ +
= ( ˆ ) − ( )
is called the ‘bootstrap error’ . Typically ∗ −
( ˆ ) can be made arbitrarily small by taking
enough resamples. We say that ‘the bootstrap
works’ if → 0. This involves more than the
continuity of the functional (·), since itself
varies with . If the bootstrap works, then
∗ − ( ) → 0as →∞
Our analysis of will proceed by subtracting,
from each of its terms, the limit (assumed to exist)
( )of ( ); in the simplest cases this limit
does not depend on .