Asymptotic Methods in Statistical Inference - Statistics Centre

50
• Asymptotic tests are generally not unique, in that
they have ‘equivalent’ modifications. Two sequences
of tests, with rejection regions and
0 and test statistics and are asymptotically
equivalent if, under , the probability of
their leading to the same conclusion tends to 1:
( ∈ and ∈ )+
0
( ∈ and ∈ ) 0 → 0 (5.2)
• Now consider AN test statistics with differing estimates
of scale. We test : = 0 vs. : 0
using
( √ )
( −
= | =
0 )
ˆ
( √ )
0 ( −
= | =
0 )
where ˆ and ˆ 0 are consistent estimates of ( 0 )
and is ( 0 2 ( 0 ) ) under. Then
→ Φ, so that (5.2) holds, e.g. the first
ˆ 0