Asymptotic Methods in Statistical Inference - Statistics Centre

103
We minimize
Z h
2 ()+2 2 () − 2() i
by minimizing the integrand pointwise: 2 − (2 −
2 2 ) is minimized over ≥ 0by =( − 2 ) + ;
thus
() =( − 2 ) +
which is symmetric and non-negative. Necessarily
0 for integrability; then also 0else ≡ 0. Thus
we can write as
à !
() = 1 − 2
2 || ≤
Thesidecondition R () =1gives =3(4) and
then R 2 () =1gives = √ 5; thus
Ã
() = 3
4 √ 5
1 − 2 5
|| ≤ √ 5
and we calculate that
" Z
√ # 2 5
() =12
− √ 2 () = 108
5 125 = 864
If has this density then 2 5 ∼ (12 2).
!