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 p(ij) = 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 H0:p(ij) = p(ji) for-all(i,j) vs. the alternative H1: p(ij) greater-than-or-equal-to p(ji), 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 p(ij)'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.