If statistics is about sorting information from error, then, arguably, half of the theory of statistical estimation is missing. Estimation theory, at its beginning, paid sampling error (also known as random error or discrepancy, as opposed to bias) little consideration, other than to state that it exists and must be overcome. Nowhere, it seems, did our statistical forebearers consider so fundamental a question as: Is it possible to estimate the amount of sampling error in a statistic from a single sample?” Once the issue did appear in the literature, it seems to have been immediately and summarily answered in the negative, without formal justification, a status that has remained unchanged and, apparently, unquestioned. In this paper, I argue that the omission of a theorem on sampling error estimation diminishes estimation theory, and that our collective blind spot in this respect represents a persistent limitation on our reason and imagination. I propose that a good start would be to restate the question as a scientific query, in unambiguous and falsifiable terms, and I provide my own effort to that end.

/pdf/the-missing-theorem-can-sampling-error-be-estimated-1lwecmqewv.pdf

The Missing Theorem: Can sampling error be estimated?

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.

/pdf/on-statistical-deficiency-why-the-test-statistic-of-the-4bxm3wtzvf.pdf

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

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