Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.3. Likelihood Ratio Tests 495

Table 8.3.1:EmpiricalαLevels for the Nominal 0.05t-Test of Example 8.3.4.

Empiricalα

μc 0 5 10 15 20

α̂ 0.0458 0.0961 0.1238 0.1294 0.1301

8.3.19 shows, the sampling error in the table is about 0.004). Note that whenμc=0

the distribution ofXis symmetric about 0 and in this case the empirical level is

close to the nominal value of 0.05.

8.3.2 Likelihood Ratio Tests for Testing Variances of Normal

Distributions

In this section, we discuss likelihood ratio tests for variances of normal distributions.

In the next example, we begin with the two sample problem.

Example 8.3.5.In Example 8.3.1, in testing the equality of the means of two
normal distributions, it was assumed that the unknown variances of the distributions
were equal. Let us now consider the problem of testing the equality of these two
unknown variances. We are given the independent random samplesX 1 ,...,Xnand
Y 1 ,...,Ymfrom the distributions, which areN(θ 1 ,θ 3 )andN(θ 2 ,θ 4 ), respectively.
We have
Ω={(θ 1 ,θ 2 ,θ 3 ,θ 4 ):−∞<θ 1 ,θ 2 <∞, 0 <θ 3 ,θ 4 <∞}.
The hypothesisH 0 :θ 3 =θ 4 , unspecified, withθ 1 andθ 2 also unspecified, is to be
tested against all alternatives. Then

ω={(θ 1 ,θ 2 ,θ 3 ,θ 4 ):−∞<θ 1 ,θ 2 <∞, 0 <θ 3 =θ 4 <∞}.

It is easy to show (see Exercise 8.3.11) that the statistic defined by Λ =L(ˆω)/L(Ω)ˆ
is a function of the statistic

F=

∑n

1

(Xi−X)^2 /(n−1)

∑m

1

(Yi−Y)^2 /(m−1)

. (8.3.9)

Ifθ 3 =θ 4 , this statisticFhas anF-distribution withn−1andm−1 degrees of

freedom. The hypothesis that (θ 1 ,θ 2 ,θ 3 ,θ 4 )∈ωis rejected if the computedF≤c 1

or if the computedF≥c 2. The constantsc 1 andc 2 are usually selected so that, if

θ 3 =θ 4 ,

P(F≤c 1 )=P(F≥c 2 )=

α 1

2

,

whereα 1 is the desired significance level of this test. The power function of this
test is derived in Exercise 8.3.10.