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## Background

The Neyman Pearson Theorem provides a method of constructing most powerful tests for simple hypotheses when the distribution of the observation is known except for the value of a single parameter. But in many cases the problem is more complex than this. In this section we will examine a general method that can be used to derive tests of hypotheses. The procedure works for simple or composite hypotheses and whether or not there are `nuisance' parameters with unknown values.

As well as thinking of as being a statement (or assertion) about a parameter , it is a set of values taken by . Similarly for . So it is appropriate to write , for example, or . The set of all possible values of is .

Let be the density function of a random variable with unknown parameter , and let be a random sample from this distribution, with observed values . The likelihood function of the sample is

It is necessary to have a clear idea of what is meant by the parameter space and that subset of it defined by the hypothesis.

Example 9..7

(a)
If is distributed as bin(n,p) and we are testing , then can be written and .
(b)
If is distributed as N(, ) where both and are unknown and we are testing , then

Now (b) is illustrated in the diagram below.

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