Asymptotic Methods in Statistical Inference - Statistics Centre

94
• When is less well-behaved, ad hoc methods are
required. Let 1 ∼ ( 1) and estimate
() = by ( ¯). The derivatives of
are unbounded so the theorem above does not
apply. But
√ ³ h ( ¯) − ()´ → (0 0 () i 2
= 2 )
and
√ ( ¯) = √ ¯ = √ ¯−
has variance 2 var h √ ¯− i , so we have equality
of the limiting and asymptotic variances if
var h ¯− i → 1. With √ ³ ¯ − ´
= ∼
(0 1) and () = h i = 22 we have that
var
∙ ¯−¸ = · var h √ i
= n h 2√ i − 2 h √ io
= n (2 √ ) − 2 (1 √ ) o
= n 2 − 1o (exactly!)
=
½∙1+ 2 + ³ −2´¸
−
∙1+ 1 + ³ −2´¸¾
= 1+ ³ −1´
as required.