Likelihood ratio tests for and against decreasing in transposition in KxKxK contingency tables

We consider testing for and against decreasing in transposition in KxKxK contingency tables. We show that when testing for exchangeability against this type of ordering, the asymptotic distribution of the likelihood ratio statistic is that of a convolution of several independent chi-bar square distributions and hence is itself a chi-bar square distribution. We provide expressions for the weighting values and we obtain the least-favorable distribution for testing for this type of ordering. Details are given for the cases K=2 and K=3. An illustrative example is included.