Asymptotic Methods in Statistical Inference - Statistics Centre

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that then √ ³ ¯ − ´ → (0
2 )where,with
= −1 (1 − ),
2 =
h
min (|| ) 2i (1 − 2) 2
=
" Z
2
(1 − 2) 2 2 () +
2 0
This reduces to 2 as → 0( →∞). Here
we use that
2 = h
||
2 i
h
≥ min (|| ) 2i
= (1− 2) 2 2 (11.1)
= 2
" Z
0
2 () + 2
so that 2 → 0as → 0. As → 12, 2 →
1 h 4 2 (0) i , by two applications of L’Hospital’s
rule.
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