Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.3. Maximum Likelihood Tests 381

whereR 0 nconverges to 0 in probability. Hence the following decision rule defines
an asymptotic levelαtest underH 0 :

RejectH 0 in favor ofH 1 ifχ^2 R≥χ^2 α(1). (6.3.20)

Example 6.3.3(Example 6.2.6, Continued).As in Example 6.2.6, letX 1 ,...,Xn

be a random sample having the common beta(θ,1) pdf (6.2.14). We use this pdf to

illustrate the three test statistics discussed above for the hypotheses

H 0 :θ=1versusH 1 :θ =1. (6.3.21)

UnderH 0 ,f(x;θ) is the uniform(0,1) pdf. Recall that̂θ=−n/

∑n
i=1logXiis the
mle ofθ. After some simplification, the value of the likelihood function at the mle
is

L(̂θ)=

(

−

∑n

i=1

logXi

)−n

exp

{

−

∑n

i=1

logXi

}

exp{n(logn−1)}.

Also,L(1) = 1. Hence the likelihood ratio test statistic is Λ = 1/L(̂θ), so that

χ^2 L=−2logΛ = 2

{

−

∑n

i=1

logXi−nlog

(

−

∑n

i=1

logXi

)

−n+nlogn

}

.

Recall that the information for this pdf isI(θ)=θ−^2. For the Wald-type test, we
would estimate this consistently bŷθ−^2. The Wald-type test simplifies to

χ^2 W=

(√

n

θ̂^2

(̂θ−1)

) 2

=n

{

1 −

1

θ̂

} 2

. (6.3.22)

Finally, for the scores-type course, recall from (6.2.15) that thel′(1) is

l′(1) =

∑n

i=1

logXi+n.

Hence the scores-type test statistic is

χ^2 R=

{∑n

i=1log√Xi+n

n

} 2

. (6.3.23)

It is easy to show that expressions (6.3.22) and (6.3.23) are the same. From Example
6.2.4, we know the exact distribution of the maximum likelihood estimate. Exercise
6.3.8 uses this distribution to obtain an exact test.

Example 6.3.4(Likelihood Tests for the Laplace Location Model).Consider the

location model

Xi=θ+ei,i=1,...,n,

where−∞<θ<∞and the random errorseis are iid each having the Laplace pdf,

(2.2.4). Technically, the Laplace distribution does not satisfy all of the regularity