Asymptotic Methods in Statistical Inference - Statistics Centre

161
• Example 2. ( )=bias. Write ˆ = ( ˆ )=
( 1 ); then
( ) = [ ( 1 )] − ( )
( ˆ ) = ˆ
[ ( ∗ 1 ∗ )] − ( ˆ )
To approximate the ³ latter we draw bootstrap
samples
∗
1
∗ ´, compute
∗ = ³ 1 ∗ ´ ∗ each time and
⎡
∗ = ⎣ 1
By the WLLN, ∗
⎤
X
∗
=1
⎦ − ˆ
→ ( ˆ )as →∞.
• There are two aspects of bootstrapping being considered
here. The first is the computation of ∗
to estimate ( ˆ ), which we use as our finitesample
measure of the performance of ³ ˆ ´.
Matters might end here if this is all we want. Or,
we can ask — and will for the rest of this lecture
— how good ( ˆ )isasanestimateof ( ):
∗
−→
→∞ ( ˆ ) ? ≈ ( )