Asymptotic Methods in Statistical Inference - Statistics Centre

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24. Examples
Example: 1 ∼ (), 1 ∼ ().
Test = by defining = + and testing =0
vs. 6= 0;thenθ =( ). Assume →∞with
( + ) → ∈ (0 1). See Theorem 7.6.3 — the
asymptotic normality of consistent sequences of local
maxima of the likelihoods continues to hold, with
I (θ) = (1− )I X (θ)+I Y (θ)
= lim 1
+ h −¨ (θ) i
where I X (θ) is the information matrix based on the
distribution of 1 ,etc.and¨ (θ) iscomputed
from all = + observations. The basic idea
is that the log-likelihood is a convex combination of
that from and that from , and its averages and
moments converge to the corresponding convex combinations;
e.g.
1
h −¨ (θ) i
"
−¨
= (θ)
+ + −¨ #
(θ)
+
→ (1 − )I X (θ)+I Y (θ)