**Date & Time:** July 11, 2012 :: 15:30-16:30

**Venue:** Ramanujan Hall

**Title:**A distribution-free test for bivariate symmetry about a line

**Speaker:** Prof. K.S. Madhava Rao,
Department of Statistics, University of Botswana, Gaborone

**Abstract:** Testing of hypotheses of symmetry in univariate and multivariate distributions in the
nonparametric set up have been dealt by many researchers over the years. In multivariate set up,
there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry,
marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and
elliptical symmetry among others. In this paper, for bivariate case, we formulate a concept of
symmetry about a straight line passing through the origin in a plane and develop a simple
distribution-free test for testing the hypothesis of symmetry about a straight line. The proposed test
is based on a measure of deviance between observed counts of bivariate samples in suitably
defined pairs of sets. The exact null distribution of the test statistics and its non-null distribution
for specified classes of alternatives are obtained and the null distribution is tabulated for sample
size from n =5up to n =30. The exact null mean and null variance and asymptotic null
distributions of the proposed test statistic are also obtained. The empirical power of the proposed
test is evaluated by simulating samples from suitable class of bivariate distributions. The empirical
findings suggest that the test performs reasonably well against various classes of asymmertic
bivariate distributions. Further, it is noted that the basic idea developed in this work can be easily
adopted to test hypotheses of exchangeability of bivariate random variables and also bivariate
symmetry about a given axis which have considered earlier by several authors.

Keywords: bivariate symmetry, count statistics, empirical power, nonparametric test. Mathematics Subject Classification: primary 62H99; secondary 62G99. …………………………………………………………………………………………………………………