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 - 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: 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 - 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
(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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