PDF of Lecture Notes - School of Mathematical Sciences

2. STATISTICAL INFERENCE
=⇒ W =
√
ni(ˆλ)(ˆλ − λ 0 )
= ˆλ − λ 0
√ˆλ/n
V = U(λ 0; x)
√
ni(λ0 )
= n(¯x − λ 0)
λ 0 ( √ n/λ 0 )
= ¯x − λ 0
√
λ0 /n
G 2 = 2(l(ˆλ) − l(λ 0 ))
= 2n(¯x log ˆλ − ˆλ) − 2n(¯x log λ 0 − λ 0 )
= 2n
(
¯x log ˆλ
)
− (ˆλ − λ 0 ) .
λ 0
Remarks
(1) It can be proved that the tests based on W, V, G 2 are asymptotically equivalent
for H 0 true.
(2) As a by-product, it follows that the null distribution of G 2 is χ 2 1. (Recall
from Theorems 2.3.1, 2.3.1 that the null distribution for W, V are both
N(0, 1)).
(3) To understand the motivation for the three tests, it is useful to consider
their relation to the log-likelihood function. See Figure 20.
We have introduced the Wald test, score test and the likelihood test for H 0 : θ = θ 0 vs.
H A : θ = θ a . These are large-sample tests in that asymptotic distributions for the test
statistic under H 0 are available. It can also be proved that the LR statistic and the
score test statistic are invariant under transformation of the parameter, but the Wald
test is not.
Each of these three tests can be inverted to give a confidence interval (region) for θ:
Wald Test
108