Show that for k 2 the likelihood ratio statistic of

Question

Show that for k = 2 the likelihood ratio statistic of Exercise 25 can be expressed in terms of the ratio of the two sample variances and that the likelihood ratio test can, therefore, be based on the F distribution.

Exercise 25

Independent random samples of sizes n₁, n₂, ... , and n_k from k normal populations with unknown means and variances are to be used to test the null hypothesis  =  =···=  against the alternative that these variances are not all equal.

(a) Show that under the null hypothesis the maximum likelihood estimates of the means μ₁ and the variances  are

(b) Using the results of part (a), show that the likelihood ratio statistic can be written as