 Solution

Ignoring a factor independent of the parameters, the likelihood function isSince every count is positive, a boundary value ${\theta }_{ij}=0$ has zero likelihood. At an interior optimum, a Lagrange multiplier for $\sum {\theta }_{ij}=1$ gives $n_{ij}/{\theta }_{ij}=\lambda$. Summing the resulting ${\theta }_{ij}=n_{ij}/\lambda$ gives $\lambda =N$, soThe log likelihood function is strictly concave on the positive simplex, because its Hessian matrix is diagonal with entries $-n_{ij}/{\theta }_{ij}^2<0$. Hence this critical point is the unique global maximum-likelihood estimate.