Asymptotic Methods in Statistical Inference - Statistics Centre

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• We often work instead with the observed value
= (x) of andthencalculatethep-value:
ˆ( ) = 0 ( )
Ã√ ( −
= 1− Φ
0 )
( 0 )
!
+ (1)
if is ³ 0 2 ( 0 )´. [The error is uniformly
(in ) (1) by Theorem 2.6.1 - how?]
• Studentization: The above derivation holds with
( 0 ) replaced by any consistent estimate. More
generally, suppose that
√ ( − 0 ) → ³ 0 2 ( 0 )´
for a (possibly vector-valued) ‘nuisance parameter’
. E.g. √ ³ ³
¯ − 0´ → 0
2´
and 2 is
a nuisance parameter if interest is on testing for
. Suppose that, when is true, ˆ → ( 0 ).
Then Slutsky’s Theorem yields the critical point
= 0 + ˆ
√
+
³
−12´