A new approach to Student and Fisher using polynomial roots
18 Pages Posted: 8 Jun 2020 Last revised: 2 Dec 2022
Date Written: December 2, 2022
We provide a new pathway to the derivation of a half-dozen statistical results under standard assumptions, including key results due to Student and Fisher. To the best of our knowledge, these are the first new derivations of these results in 75 years. Our work links two seemingly disparate literatures (the geometrical properties of polynomial roots in the complex plane and the mathematical statistics of Student and Fisher). Given a random sample of real numbers, we insert the numbers into a polynomial as its coefficients and we extract the polynomial’s roots. We develop a dual space analysis using the location in the complex plane of these roots relative to the location around the unit circle of the roots of the cyclotomic equation z^N − 1 = 0. Multiple applications of Pythagoras’ theorem lead to a canonical orthogonal transformation and decomposition of the sample variance of the original sample of real numbers. This variance decomposition allows for the immediate proof of the statistical results. A surprising consequence of our complex mathematics is a set of algebraic examples so simple that they contribute notably to the understanding of statistical methodology. We also discuss several promising directions for future research using our dual space method.
Keywords: Dual space, Complex plane, Roots of polynomials, Orthogonal decomposition, Sample variance, Student-t
JEL Classification: C1, C46, G1
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