Asymptotic Methods in Statistical Inference - Statistics Centre

160
• Example 1. =3,( 1 2 3 )=(8 3 5), =
median of , ˆ = 5. A bootstrap sample might
be X ∗ =(8 8 5), obtained by randomly drawing
three values, with replacement, from (8 3 5). In
this case ∗ =8. After repetitions of the procedureweapproximate
( ˆ )(stillincase(i)
above) by
∗ = 1
X
=1
X
³ √
³
∗
− ˆ ´
≤ ´
= 1 ³ √ ³
3
∗
− 5´
≤ ´
=1
Thus ∗ ∼ 1 ³ ( ˆ )´
, hence → ( ˆ )
as →∞.
• In this example ( ˆ ) can be computed exactly
— there are only 3 3 = 27 possible resamples, with
medians ∗ =(8 3 5) occurring with probabilities
(under ˆ )of(7 7 13) 27. These are the
probabilities of the three possible values of
√
3(
∗ − 5), and then one easily calculates
( ˆ )= ˆ
h√
3(
∗ − 5) ≤ i