Asymptotic Methods in Statistical Inference - Statistics Centre

207
• Multiparameter inferences. Test (simple null)
: θ = θ 0 vs. : θ 6= θ 0 where θ is × 1
and 1 ∼ ( θ). The multiparameter
versions of the tests given above are as follows.
(Note: all evaluations from now on will assume
that is true.)
— Wald’s test:
√
³ˆθ − θ 0´ →
³
0 I −1 ( 0 )´
⇒
2 = ³ˆθ − θ 0´0
I (0 ) ³ˆθ − θ 0´ →
2
and is rejected if 2 2 (). The corresponding
confidence region is the -dimensional
ellipsoid
½
θ ³ˆθ ¾
0 | − θ 0´0
Î
³ˆθ − θ 0´
≤
2 ()
where Î = I ³ˆθ ´
or −
1
¨
³ˆθ ´.
— Likelihood ratio test:
2∆ =2 ³
³ˆθ ´
− (θ 0 )´ →
2