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