Asymptotic Methods in Statistical Inference - Statistics Centre

142
• Proofoftheorem: Weusethe‘projectionmethod’,
by which we approximate = √ ( − ) bya
sum ∗ of i.i.d.s, apply the CLT to ,thenshow
∗
that the approximation is good enough that the
end result applies to as well. Since the best
(minimum mse) forecast of a r.v. from another
is ... (what?), it is reasonable to conjecture that
the ‘best’ approximation is the ‘projection on the
observations’
∗ =
X
=1
h √ ( − ) |
i
Claim 1: [( − ) | ]=()( 1 ( ) − ),
so that by the CLT, ∗ → ³ 0 2 1´. 2 (The
mean 0 and variance 2 1 2 are exact for all .)
Claim 2: [ ]= ∗ 2 1 2 . Note this also
=lim[ ]=[], ∗ so that
h ( − ∗ ) 2i = [ − ∗ ] → 0
and − ∗ → 0 in quadratic mean, hence in
probability. Thus
= ∗ +( − ) ∗ → ³ 0 2 1
2
by Slutsky’s Theorem, completing the proof.
´