What means "CDF of one group does not cross the CDF of the other" from the dunn.test description? From the dunn.test manual: "CDF of one group does not cross the CDF of the other". What this means? I am not a statistician, but I want to understand. What I found using a Google search not helped me very much :(.
 A: CDF's are Cumulative Distribution Functions, i.e. they tell you about the probability that some value $X$ is less or equal then $x$, $F(x) = P(X \leq x)$.
Below you can see the example of crossing and non crossing CDF's:
set.seed(123)

n <- 1e3
x <- sort(rnorm(n, 0, 1))
y <- sort(rnorm(n, 1, 1))
z <- sort(rnorm(n, .5, 2))

plot(x, (1:n)/n, type="l", xlim=c(-4, 4), ylab="F(x)")
lines(y, (1:n)/n, col="red")
lines(z, (1:n)/n, col="green")


On the y-axis there are cumulative probabilities and on the x-axis values of the variables. Lines black (x) and red (y) do not cross, while green (z) crosses both black and white. What this plot tells you is that values of x and y go in parallel, because the only difference between them is that their mean value is different. Distribution of z is different because it has different standard deviation, so its CDF also differs.
Now, if you read again the manual of dunn.test function in R (bold font by myself):

The interpretation of stochastic dominance requires an assumption that
  the CDF of one group does not cross the CDF of the other. (...) The
  null hypothesis for each pairwise comparison is that the probability
  of observing a randomly selected value from the first group that is
  larger than a randomly selected value from the second group equals one
  half; this null hypothesis corresponds to that of the
  Wilcoxon-Mann-Whitney rank-sum test. Like the rank-sum test, if the
  data can be assumed to be continuous, and the distributions are
  assumed identical except for a difference in location, Dunn’s test may
  be understood as a test for median difference

...you can see the assumption of Dunn test is that the distributions are the same and differ just by location (i.e. mean). This is exactly what is illustrated on the plot I pasted, where x and y have different mean but z has also different standard deviation and so its distribution has different shape.
A: CDF is a monotonously increasing curve with y-values going from 0 to 1 and x values going through the entire domain. If you draw two curves on the same graph, they may or may not cross each other. They'll definite connect at y=0 and 1.
