[R] Comparing distributions

[Ralf B]

I am trying to do something in R and would appreciate a push into the
right direction. I hope some of you experts can help.

I have two distributions obtrained from 10000 datapoints each (about
10000 datapoints each, non-normal with multi-model shape (when
eye-balling densities) but other then that I know little about its
distribution). When plotting the two distributions together I can see
that the two densities are alike with a certain distance to each other
(e.g. 50 units on the X axis). I tried to plot a simplified picture of
the density plot below:




|
|                                                         *
|                                                      *     *
|                                                   *    +   *
|                                              *     +     +  *
|                     *        +           *   +            +  *
|                 *        +*     +   *  +                   + *
|              *       +       *     +                           +*
|           *       +                                               +*
|        *       +                                                    +*
|     *      +                                                          + *
|  *      +                                                               + *
|___________________________________________________________________


What I would like to do is to formally test their similarity or
otherwise measure it more reliably than just showing and discussing a
plot. Is there a general approach other then using a Mann-Whitney test
which is very strict and seems to assume a perfect match. Is there a
test that takes in a certain 'band' (e.g. 50,100, 150 units on X) or
are there any other similarity measures that could give me a statistic
about how close these two distributions are to each other ? All I can
say from eye-balling is that they seem to follow each other and it
appears that one distribution is shifted by a amount from the other.
Any ideas?

Ralf