Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

378 Maximum Likelihood Methods

Note that under the null hypothesis,H 0 ,thestatistic(2/θ 0 )

∑n

i=1Xihas aχ

2

distribution with 2ndegrees of freedom. Based on this, the following decision rule

results in a levelαtest:

RejectH 0 if (2/θ 0 )

∑n

i=1Xi≤χ

2

1 −α/ 2 (2n)or(2/θ^0 )

∑n

i=1Xi≥χ

2

α/ 2 (2n),(6.3.5)

whereχ^21 −α/ 2 (2n)isthelowerα/2 quantile of aχ^2 distribution with 2ndegrees

of freedom andχ^2 α/ 2 (2n) is the upperα/2 quantile of aχ^2 distribution with 2n
degrees of freedom. Other choices ofc 1 andc 2 can be made, but these are usually
the choices used in practice. Exercise 6.3.2 investigates the power curve for this
test.

g(t)

t

c 1 c 2

c

Figure 6.3.1:Plot for Example 6.3.1, showing that the functiong(t)≤cif and
only ift≤c 1 ort≥c 2.

Example 6.3.2(Likelihood Ratio Test for the Mean of a Normal pdf).Consider
a random sampleX 1 ,X 2 ,...,Xnfrom aN(θ, σ^2 ) distribution where−∞<θ<∞
andσ^2 >0 is known. Consider the hypotheses

H 0 :θ=θ 0 versusH 1 : θ =θ 0 ,

whereθ 0 is specified. The likelihood function is

L(θ)=

(

1

2 πσ^2

)n/ 2

exp

{

−(2σ^2 )−^1

∑n

i=1

(xi−θ)^2

}

=

(

1

2 πσ^2

)n/ 2

exp

{

−(2σ^2 )−^1

∑n

i=1

(xi−x)^2

}

exp{−(2σ^2 )−^1 n(x−θ)^2 }.

Of course, in Ω ={θ:−∞<θ<∞},themleisθ̂=Xand thus

Λ=

L(θ 0 )

L(̂θ)

=exp{−(2σ^2 )−^1 n(X−θ 0 )^2 }.