A new approach to Student and Fisher using polynomial roots

18 Pages Posted: 8 Jun 2020 Last revised: 2 Dec 2022

See all articles by Timothy Falcon Crack

Timothy Falcon Crack

University of Otago - Department of Accountancy and Finance

Michael Osborne

University of Sussex Business School

Malcolm Crack

University of Otago, Students

Mark Osborne

affiliation not provided to SSRN

Date Written: December 2, 2022

Abstract

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

Suggested Citation

Crack, Timothy Falcon and Osborne, Michael J. and Crack, Malcolm and Osborne, Mark, A new approach to Student and Fisher using polynomial roots (December 2, 2022). Available at SSRN: https://ssrn.com/abstract=3598613 or http://dx.doi.org/10.2139/ssrn.3598613

Timothy Falcon Crack (Contact Author)

University of Otago - Department of Accountancy and Finance ( email )

Dunedin
New Zealand

Michael J. Osborne

University of Sussex Business School ( email )

Jubilee Building
University of Sussex, Falmer
Brighton, Sussex BNI 9SL
United Kingdom

Malcolm Crack

University of Otago, Students ( email )

PO Box 56
Dunedin, Otago
New Zealand

Mark Osborne

affiliation not provided to SSRN

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