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

The random variate m is, in combinatorics, a basis for comparing permutations, as well as the solution to a centuries-old riddle involving the mishandling of hats. In statistics, m is the test statistic for a disused null hypothesis statistical test (NHST) of association, the matching method. In this paper, I show that the matching method has an absolute and relatively low limit on its statistical power. I do so first by reinterpreting Rae’s theorem, which describes the joint distributions of m with several rank correlation statistics under a true null. I then derive this property solely from m’s unconditional sampling distribution, on which basis I develop the concept of a deficient statistic: a statistic that is insufficient and inconsistent and inefficient with respect to its parameter. Finally, I demonstrate an application for m that makes use of its deficiency to qualify the sampling error in a jointly estimated sample correlation.

💡 Research Summary

The paper investigates the long‑neglected test statistic m that underlies the classic “matching method” for testing association between two variables. In combinatorial terms, m counts the number of matches (fixed points) between two random permutations; this same quantity appears in the famous hat‑check problem. Historically, statisticians have used m as the test statistic for a null‑hypothesis significance test (NHST) of independence, but the method has fallen out of favor. The author explains why, by showing that m possesses an intrinsic statistical deficiency that caps its power at a very low absolute level, regardless of sample size.

The analysis proceeds in three stages. First, the author revisits Rae’s 1975 theorem, which characterizes the joint distribution of m together with several rank‑correlation statistics (Spearman’s ρ, Kendall’s τ, etc.) under the null hypothesis of independence. By re‑deriving Rae’s results, the paper demonstrates that, under the null, m follows a distribution that is essentially Poisson with mean 1, giving E