Dual Space Arguments Using Polynomial Roots in the Complex Plane: A Novel Approach to Deriving Key Statistical Results
23 Pages Posted: 8 Jun 2020
Date Written: January 30, 2020
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: Suggested Citation