Asymptotic Methods in Statistical Inference - Statistics Centre

192
The Fisher information about is
() = 1 h − 00 () i = 2 ³ 2+ 2´
and so the information about = () (with
() =1 and 0 () =− 2 )is
() = () h 0 () i 2 2+ 2
=
Thus √ (ˆ − ) → ³ 0 −1 () = 2 ³ 2+ 2´´.
Two other estimates of (= ) are 1 = ¯
(with √ ( 1 − ) → ³ 0 2 2´) and 2 = :
2
√
³
2 − 2´ →
³
0 2
4´
⇒ √ ( − ) → ³ 0 2 2´
Of course each is asymptotically more highly varied
than the MLE, but consider =(1− ) 1 +
2 .
Since 1 and 2 are independent, we have (how?)
√ ( − ) =(1− ) √ ( 1 − )+ √ ( 2 − )
→
Ã
0 (1 − ) 2 2
2 + 22
2
!