Dual Space Arguments Using Polynomial Roots in the Complex Plane: A Novel Approach to Deriving Key Statistical Results

23 Pages Posted: 8 Jun 2020

See all articles by Timothy Falcon Crack

Timothy Falcon Crack

University of Otago

Michael Osborne

University of Sussex Business School

Malcolm Crack

University of Otago, Students

Mark Osborne

affiliation not provided to SSRN

Date Written: January 30, 2020

Abstract

We present a canonical orthogonal decomposition of sample variance and its applications. Surprisingly, our decomposition arises naturally from a novel dual space argument using polynomial roots in the complex plane. Linking these two seemingly disparate literatures yields a new pathway to the derivation of key statistical results under standard assumptions. These results include the chi-squared distribution of the scaled sample variance, the loss of one degree of freedom (relative to sample size) in the sample variance, the distribution of Snedecor’s F-test of differences in dispersion, the independence of the sample mean and sample variance, and the distribution of the one-sample Student-t test of the mean. We suggest several promising directions for future research using our dual space method.

Keywords: Degrees of freedom, Orthogonal decomposition, Sample variance, Snedecor’s F-test, Student-t test

JEL Classification: C1, C46, G1

Suggested Citation

Crack, Timothy Falcon and Osborne, Michael J. and Crack, Malcolm and Osborne, Mark, Dual Space Arguments Using Polynomial Roots in the Complex Plane: A Novel Approach to Deriving Key Statistical Results (January 30, 2020). 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 ( email )

P.O. Box 56
Dunedin, Otago 9010
New Zealand

Michael J. Osborne

University of Sussex Business School ( email )

Jubilee Building
University of Sussex, Falmer
Brighton, Sussex BNI 9SL
United Kingdom
(44)1273872687 (Phone)

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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