Robert............I would be most interested in having a copy of the paper..........thank you..............dale
Dale N. Glaser, Ph.D. Pacific Science & Engineering Group 6310 Greenwich Drive; Suite 200 San Diego, CA 92122 Phone: (858) 535-1661 Fax: (858) 535-1665 http://www.pacific-science.com -----Original Message----- From: Robert J. MacG. Dawson [mailto:[EMAIL PROTECTED]] Sent: Friday, January 03, 2003 5:59 AM To: Francois Bergeret Cc: [EMAIL PROTECTED] Subject: Re: Question on Wilcoxon Test 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 I showed some years back (unpublished) that for well-enough behaved distributions (density analytic, with all moments) that symmetry is a *necessary* condition here; given any asymmetric B-F family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/ . ================================================================= . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
