Next: Asymptotic Distribution of Up: Likelihood Ratio Tests Previous: The Likelihood Ratio Test   Contents

## Some Examples

Example 9..8
Let have a normal distribution with unknown mean and known variance . Suppose we have a random sample from this distribution and wish to test against . Now

Rather than we have here or more briefly, .

Now is obtained by replacing in the above by its mle, . So

Also has only one value, obtained by replacing by and by . So

Thus, on simplification
 (9.7)

Intuitively, we would expect that values of close to 3 support the hypothesis and it can be seen that in this case is close to . Values of far from lead to close to . We need to find the critical value to satisfy (3.6). That is, we need to know the distribution of . From (3.7), using random variables instead of observed values, we have

which is the square of a variate and therefore is distributed as . For , the critical region is , or alternatively

The relationship between the critical region for and the critical region for is shown in the diagram below.

Example 9..9

Given is a random sample from a distribution, where is unknown, derive the LR test of against .

Now the parameter space is

and that restricted by the hypothesis is

We note that there are unknown parameters here, and the likelihood of the sample,
can be written as
 (9.8)

Now the mle's of and are

and is obtained by substituting these for and in (3.8). This gives
 (9.9)

Now is obtained by substituting for in (3.8) and replacing by its MLE where is known. This is , say.
Thus

So becomes

Taking th powers of both sides and writing as , we have
 (9.10)

Recalling that is the observed value of a random variable with a range space , and that the critical region is of the form , we would like to find a function of (or of whose probability distribution we recognize. Now

Also, so where . So, expressing the denominator of (3.10) in terms of random variables we have

where is a random variable with a distribution on degrees of freedom. Considering range spaces, the relationship between (or ) and is a strictly decreasing one, and a critical region of the form is equivalent to a CR of the form , as indicated in the diagram below.

That is, the critical region is of the form where is obtained from tables, using degrees of freedom, and the appropriate significance level.

Example 9..10

Given the random sample from a distribution, derive the LR test of the hypothesis , where is unknown, against .

The parameter space, and restricted parameter space are

is given by (3.8) in Example 3.2, and is given by (3.9). To find we replace by and by the mle, . So

So

Again, we would like to express as a function of a random variable whose distribution we know. Denoting by , the random variable , whose observed value this is, has a distribution with parameter . So we have

and the relationship between the range spaces of and is shown in the diagram below

A critical region of the form corresponds to the pair of intervals, . So for a size- test, is rejected if

[This of course is the familiar intuitive test for this problem.]

Next: Asymptotic Distribution of Up: Likelihood Ratio Tests Previous: The Likelihood Ratio Test   Contents
Bob Murison 2000-10-31