The Maximum-Likelihood Estimate for Contingency Tables with Zero Diagonal

Under a reasonable hypothesis about the distribution of trade shipments among a group of K countries, or of non-self-directed acts among any group of K individuals, the likelihood of obtaining a given table of observed frequencies n ij is where n i = n i 1 + ··· + n iK and m i = n 1i + ··· +n Ki are the total number of shipments made and received, respectively, by country i, N is the total number of shipments altogether, and P i and Q,i are the theoretical tendencies of country i to ship and to receive shipment respectively and satisfy ΣP i = ΣQ i = 1. In the attempt to test the correctness of the hypothesis, a critical problem is that of finding the maximum-likelihood estimates of the P's and Q's, that is, the values making L as large as possible. We give all solutions and show that, for the usual table of observations, the solution is unique and amounts to finding the value of α for which K − 2 – R 1 – … – R K is 0- where R i = - and setting