Asymptotic Methods in Statistical Inference - Statistics Centre

173
• Assume:
(C1) Identifiability: 1 6= 2 ⇒ the distributions
1 2 are distinct. (This is violated if, e.g.,
1 ∼ ( 1) but only = || is observed,
then ± both lead to the same distributions
of : () =[ ( − )+ ( + )] (
0).)
(C2) The parameter space is an open interval:
Ω = ³ ¯´
(possibly infinite). (Ensures that maxima
of the likelihood function correspond to critical
points, assuming differentiability.)
(C3) The observations 1 arei.i.d.with
density or p.m.f. (). (Then () = P log ( ).)
(C4) The support = { | () 0} is independent
of . (This excludes cases like ∼
(0), which can be handled using other techniques.)
(C5) For all ∈ , () isdifferentiable w.r.t. ,
with derivative 0 (). (Then maxima of satisfy
the likelihood equation 0 () = P 0( )
( ) =0.)