Asymptotic Methods in Statistical Inference - Statistics Centre

181
• Example: 1 the indicators of Bernoulli
trials: = ( = 1) ∈ (0 1). Then ()
= (1 − ) 1− for ∈ {0 1} with
log () = log +(1− )log(1− )
log ()
= − 1 −
1 −
2 log ()
2 = −
2 −
"
2 #
log
()
2
1 −
(1 − ) 2
−1
=
(1 − ) ;
hence −1 () = (1 − ) =asym. var.of √ ³ˆ − ´
where (you should verify) ˆ = P . (Note
that (C2) — Ω an open interval — excludes the values
=0 1. But in these cases the MLEs exist
and are ˆ =0 1, with −1 () =0.)