Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / O / O4 / b / Solution

Under the product constraint the log-likelihood separates:Maximizing the two simplex problems independently gives ${\hat{\alpha }}_i=n_{i.}/N$ and ${\hat{\beta }}_j=n_{.j}/N$. HenceThe fitted expected cell counts under independence are consequently $N{\hat{\theta }}_{ij}^0=n_{i.}{n}_{.j}/N$.