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The random variate m is, in combinatorics, a basis for comparingpermutations, as well as the solution to a centuries-old riddle involving themishandling of hats. In statistics, m is the test statistic for a disused nullhypothesis statistical test (NHST) of association, the matching method. In thispaper, I show that the matching method has an absolute and relatively low limiton its statistical power. I do so first by reinterpreting Rae's theorem, whichdescribes the joint distributions of m with several rank correlation statisticsunder a true null. I then derive this property solely from m's unconditionalsampling distribution, on which basis I develop the concept of a deficientstatistic: a statistic that is insufficient and inconsistent and inefficientwith respect to its parameter. Finally, I demonstrate an application for m thatmakes use of its deficiency to qualify the sampling error in a jointlyestimated sample correlation.

On statistical deficiency: Why the test statistic of the matching method is hopelessly underpowered and uniquely informative