380 Maximum Likelihood MethodswhereR∗n→0, in probability. By Theorems 5.2.4 and 6.2.2, the first term on the
right side of the above equation converges in distribution to aχ^2 -distribution with
one degree of freedom.Define the test statisticχ^2 L=−2 log Λ. For the hypotheses (6.3.1), this theorem
suggests the decision ruleRejectH 0 in favor ofH 1 ifχ^2 L≥χ^2 α(1). (6.3.12)By the last theorem, this test has asymptotic levelα. If we cannot obtain the test
statistic or its distribution in closed form, we can use this asymptotic test.
Besides the likelihood ratio test, in practice two other likelihood-related tests
are employed. A natural test statistic is based on the asymptotic distribution of̂θ.
Consider the statistic
χ^2 W={√
nI(θ̂)(θ̂−θ 0 )} 2. (6.3.13)
BecauseI(θ) is a continuous function,I(θ̂)→I(θ 0 ) in probability under the null
hypothesis, (6.3.1). It follows, underH 0 ,thatχ^2 Whas an asymptoticχ^2 -distribution
with one degree of freedom. This suggests the decision rule
RejectH 0 in favor ofH 1 ifχ^2 W≥χ^2 α(1). (6.3.14)As with the test based onχ^2 L, this test has asymptotic levelα. Actually, the
relationship between the two test statistics is strong, because as equation (6.3.11)
shows, underH 0 ,
χ^2 W−χ^2 L
P
→ 0. (6.3.15)The test (6.3.14) is often referred to as aWald-type test, after Abraham Wald,
who was a prominent statistician of the 20th century.
The third test is called ascores-type test, which is often referred to as Rao’s
score test, after another prominent statistician, C. R. Rao. Thescoresare the
components of the vector
S(θ)=(
∂logf(X 1 ;θ)
∂θ,...,∂logf(Xn;θ)
∂θ)′. (6.3.16)
In our notation, we have
1
√
nl′(θ 0 )=1
√
n∑ni=1∂logf(Xi;θ 0 )
∂θ. (6.3.17)
Define the statistic
χ^2 R=(
l′(θ 0 )
√
nI(θ 0 )) 2. (6.3.18)
UnderH 0 , it follows from expression (6.3.10) that
χ^2 R=χ^2 W+R 0 n, (6.3.19)