  We discuss several tests for whether a given set of independent and
identically distributed (i.i.d.) draws does not come from a specified
probability density function. The most commonly used are Kolmogorov-Smirnov
tests, particularly Kuiper's variant, which focus on discrepancies between the
cumulative distribution function for the specified probability density and the
empirical cumulative distribution function for the given set of i.i.d. draws.
Unfortunately, variations in the probability density function often get
smoothed over in the cumulative distribution function, making it difficult to
detect discrepancies in regions where the probability density is small in
comparison with its values in surrounding regions. We discuss tests without
this deficiency, complementing the classical methods. The tests of the present
paper are based on the plain fact that it is unlikely to draw a random number
whose probability is small, provided that the draw is taken from the same
distribution used in calculating the probability (thus, if we draw a random
number whose probability is small, then we can be confident that we did not
draw the number from the same distribution used in calculating the
probability).
