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In the first two chapters we have examined the mathematical background for empirical
testing of PRNs. In this chapter, we will pass to numerical practice. We will develop a class
of empirical
tests for PRNs. We will also give hints for the design, implementation and
interpretation of these tests. As the title of the chapter suggests, we will focus our attention
on tests that interpret and complete our definition of iud. PRNs, Definition
1.8 in Chapter 2. Let us recall this definition:
As it has been explained in the last chapter every test amounts to calculating a numerical
property of sequences of PRNs which is assessed by marking a set of unwanted values as
the
critical region. Sequences that lead to results within that
critical region are rejected by the test. The numerical property we are going to
discuss in this chapter is related to the distance of the empirical distribution function of
a sample of N s-tuples of PRNs to the uniform distribution on 
. We
interpret the above
'behaves like a realization' by 'has an empirical distribution function close to the
distribution function of a random variable distributed uniformly on '.
There exist simulation problems which will yield desired results even if the empirical distribution function is far from the theoretical one, as well as there exist problems which will yield bad results although the test has been passed by the numbers.
Within this setup the PRNs are always thought to be used in the place of s-dimensional,
uniformly
distributed random variables. We will use the notation
for independent uniformly distributed random variables
and
for uniformly distributed random vectors
in the mathematical model from which we derive the distribution of the test statistic. The
are simply replaced by the actual PRNs, denoted by 
, in the evaluation of the
test for a
generator. The test statistic then becomes a PRN itself.
In order to get a practically computable quantity, we will have to make some simplifications and thereby introduce arbitrariness in the procedure. Every such step should be made with the application of the PRNs in view. If the whole test is similar in structure to the simulation problem for which the PRNs are used, confidence in the above mentioned correlation between results of the test and the simulation will be enlarged.
The mathematical treatment of the test statistic will be given in Chapter 5. An informal description will be given right in the next section.