Theses and Dissertations

ORCID

https://orcid.org/0009-0006-6735-0496

Advisor

Patil, Prakash

Committee Member

Sepehrifar, Mohammad

Committee Member

Woody, Jonathan

Committee Member

Wu, Tung-Lung

Committee Member

Zhou, Qian

Date of Degree

5-15-2026

Original embargo terms

Visible MSU Only 1 year

Document Type

Dissertation - Campus Access Only

Major

Mathematical Sciences

Degree Name

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

Due to the importance of the symmetry assumption, several contributions in the statistics literature propose various approaches to ascertain the validity of the assumption or characterize any deviations from this assumption. Classical methods characterize univariate asymmetry using quantities that are based on variants of measures of skewness. However, measures of skewness are particularly sensitive to the tail behavior of a curve, whereas symmetry is an overall distributional property that depends on the shape of the entire curve. As a result, the coefficients of skewness are ineffective in calibrating the asymmetry in the density curves. The concept of symmetry extends naturally from univariate distributions to multivariate settings where it underpins key notions such as radial symmetry, marginal symmetry, joint symmetry, conditional symmetry and argument symmetry. Similarly, several approaches have been proposed to quantify asymmetry or test for symmetry in bivariate distributions. Particularly in the context of argument symmetry, existing methods tend to focus either on local measures of asymmetry or, when global measures are considered, yield quantities that lack clear interpretability. The present work presents a necessary condition for a bivariate probability density function to be symmetric in its arguments and develops a test of bivariate argument symmetry based on it. Further, it provides a new necessary and sufficient condition of bivariate argument symmetry and uses that to measure argument asymmetry in a continuous bivariate density function on the scale of 0 to 1. In the process, it explores the relationship between central symmetry and argument symmetry of a continuous bivariate density function.

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