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## Asymptotic Distribution of

The distribution of some function of can't always be found as readily as in the previous examples. If is large and certain conditions are satisfied, there is an approximation to the distribution of that is satisfactory in most large-sample applications of the test. We state without proof the following theorem.

Theorem 9..2

Under the proper regularity conditions on , the random variable is distributed asymptotically as chi-square. The number of degrees of freedom is equal to the difference between the number of independent parameters in and .

[Note that in Example 9.3.1 the distribution of was exactly .]

Example 9..11
(Test for equality of several variances.)

The hypothesis of equality of variances in two normal distributions is tested using the -test. We will now derive a test for the -sample case by the likelihood ratio procedure. Consider independent samples

That is, we have observations .

We wish to test the hypothesis

against the alternative that the are not all the same. Let . Now the p.d.f. of the random variable is

So the likelihood function of the samples above is
 (9.11)

The whole parameter space and restricted parameter space are given by

The log of the likelihood is

using for .

To find we need the MLE's of the parameters .

 (9.12)

 (9.13)

Equating (9.12) and (9.12) to zero and solving we obtain

 (9.14) (9.15)

Substituting these in (9.11) we obtain

 (9.16)

Now in the restricted parameter space there are parameters, and . So we need to find the mle's of these parameters. The likelihood function now is (putting , all )

 (9.17)

and
 (9.18)

 (9.19)

Equating (9.18) and (9.19) to zero and solving we obtain

 (9.20) (9.21)

Substituting (9.21) and (9.21) into (9.17) we obtain
 (9.22)

So
 (9.23)

Now, using Theorem 9.3.4, the distribution of is asymptotically . To determine the number of degrees of freedom we note that the number of parameters in is and in is . Hence the number of degrees of freedom is . Thus
 (9.24)

is distributed approximately .

Bartlett (1937) modified this statistic by using unbiased estimates of and instead of MLE's. That is, he used and as divisors, so the statistic becomes

where

[Investigate the form of this when .]

A better approximation still is obtained using as the statistic where the constant is defined by

and this statistic is commonly referred to as Bartlett's statistic for testing homogeneity of variances. That is,
 (9.25)

is distributed approximately as under the hypothesis . The approximation is not very good for small .

Next: ASSIGNMENTS Up: Likelihood Ratio Tests Previous: Some Examples   Contents
Bob Murison 2000-10-31