Likelihood ratio tests for bivariate symmetry against ordered alternatives in a square contingency table

Let (X1, X2) be a bivariate random variable of the discrete type with joint probability density function pij = pr[X1 = i, X2 = j], i, J = 1, ..., k. Based on a random sample from this distribution, we discuss the properties of the likelihood ratio test of the null hypothesis of bivariate symmetry Ho: pij = pji [for all](i, j) vs. the alternative H1: pij [greater-or-equal, slanted] pji, [for all]i> j, in a square contingency table. This is a categorised version of the classical one-sided matched pairs problem. This test is asymptotically distribution-free. We also consider the problem of testing H1 as a null hypothesis against the alternative H2 of no restriction on pij's. The asymptotic null distributions of the test statistics are found to be of the chi-bar square type. Finally, we analyse a data set to demonstrate the use of the proposed tests.