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

Abstract

Let ( X 1, X 2) be a bivariate random variable of the discrete type with joint probability density function p ij = pr[ X 1 = i, X 2 = 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 H o : p ij = p ji ∀( i, j) vs. the alternative H 1: p ij ⩾ p ji , ∀ 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 H 1 as a null hypothesis against the alternative H 2 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.