Statistics PhD Defense - 03/04/26

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, coefficients of skewness are ineffective in calibrating 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.

PhD Advisor:

Dr. Prakash Patil