Asymptotic Methods in Statistical Inference - Statistics Centre

217
• If instead of testing = − = 0 one tests
= =1,both2∆ and 2 are unchanged.
But starting with
⎛ ⎛ ⎞⎞
√ ³ ¯ − ´
→ ⎝0 ⎝ 1−
0
⎠⎠
¯ −
0
then using the delta method to get the null distribution
of √ (ˆ − 1) = ¯¯ − 1, and forming
Wald’s statistic, gives
à ! 2 ¯
˜ 2 = ˆ 1 (1 − )
¯ − 1
for a consistent estimate ˆ 1 . For the original null
hypothesis (recall (24.1))
2 = (1 − ) ³ ¯ − ¯´2
ˆ 2
for a consistent estimate ˆ 2 . The ratio between
the two is
˜ 2
2
= ˆ 1ˆ 2
¯ 2
→ 1
so that the two statistics are asymptotically equal.
Whether or not Wald’s test remains the same