The WMW test is, generally, a test of the ``stochastically greater''
relation: X S> Y if P(X>Y) > 0.5. This does not generally imply (and
is not implied by) an inequality on the medians, but for some restricted
families of distributions it does/is.
Potthoff showed (1963, Ann.Math.Stat) that the WMW is a test for the
median between any two symmetric distributions. It's well-known, going
back to the original inventors, that it's a test for the median within
any family of shift alternatives f(x) = f_0(x+c). Log transformation
[which preserves the WMW statistic] shows that within any family of
scale alternatives f(x) = k f_0(kx) the same thing holds.
The next most general situation, I suppose, is the Behrens-Fisher case
of arbitrary linear transformations. I showed some years back
(unpublished) that for well-enough behaved distributions (density
analytic, with all moments) that symmetry is a *necessary* condition
here; given any asymmetric B-F family of distributions obeying these
conditions there are three RV's X,Y,Z in the family for which the S>
relation is intransitive:
X S> Y S> Z S> X
and any test for the median must be transitive. (This can be made
rigorous in terms of asymptotic behavior; if there is intransitivity
then there are pairs of distributions in which alpha -> 1 and n ->
infinity.)
Analyticity cannot be weakened here. If we take a symmetric
distribution and rescale the bottom half, the WMW will be a test for the
median of the corresponding B-F family. Usually the resulting
distributions do not even have continuous density, but... if we apply
this to a bimodal distribution with an "exp(-x^2)" flat point at the
median, the resulting thing is infinitely- differentiable, but
asymmetric.
I do not know if the existence of moments is needed or just a monument
to my lack of skill.
Finally, practical examples are likely to be rare; I can show that even
with hand-cooked nonrandom data the WMW will not actually exhibit
intransitivity at the 5% significance level with N<48, whereas with
random samples from a typical B-F family of asymmetric distributions
[shifted exponential] one of the three tests (done at the 5% level) will
have power < 50% for N<800.
I can send a DVI or paper copy to anybody interested.
-Robert Dawson
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
