Next: Neyman Pearson Theorem Up: Evaluation of and Construction Previous: Unbiased and Consistent Tests   Contents

## Certain Best Tests

When and are both simple, the error sizes and are uniquely defined. In this section we require that both the null hypothesis and alternative hypothesis are simple, so that in effect, the parameter space is a set consisting of exactly points. We will define a best test for testing against , and in 3.2.3 we will prove a Theorem that provides a method for determining a best test.

Let denote the density function of a random variable . Let denote a random sample from this distribution and consider the simple hypothesis
: and the simple alternative : . So

One repetition of the experiment will result in a particular n-tuple, ( ). Consider a set , which is a collection of n-tuples having size that is, has the property that

It follows that can be thought of as a critical region for the test. Specifically, if the observed n-tuple falls in our pre-selected , we will reject . However, if were true, then intuitively the `best' critical region would be the one having the highest probability of containing . Formalizing this notion, we have the following definition.

Definition 9..5
C is called the best critical region, (BCR) of size for testing the simple against the simple if,

(a)
(b)
for every other (of size ).
This definition can be stated in terms of power. Suppose that there is one of these subsets, say C, such that when is true, the power of the test associated with C is at least as great as the power of the test associated with each other .

Definition 9..6
A test of the simple hypothesis versus the simple alternative that has the smallest (or equivalently, the largest ) among tests with no larger is called most powerful.

Example 9..4
Suppose . Let denote the probability function of . Consider , . The table below gives the values of , and for .

 x 0 1 2 3 4 5 f(x;) f(x;) f(x;)/f(x;) 32

Using to test against , we shall first assign significance level and want a best critical region of this size. Now and are possible critical regions and there is no other subset with . So either or is the best critical region for this . If we use then P()1/1024 and

P(rejecting is true) P(rejecting is true),
an unacceptable situation. On the other hand, if we use then and
P(rejecting is true) P(rejecting is true),
a much more desirable state of affairs. So is the best critical region of size for testing against . It should be noted that, in this problem, the best critical region, C, is found by including in C the point (or points) at which is small in comparison with . This suggests that in general, the ratio provides a tool by which to find a best critical region for a certain given value of .

The theorem in the following section provides the methodology for deriving the most powerful test for testing simple against simple .

Next: Neyman Pearson Theorem Up: Evaluation of and Construction Previous: Unbiased and Consistent Tests   Contents
Bob Murison 2000-10-31