 Solution

ID: 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:

ℓ (α, β) = \sum_{i} n_{i .} lo g α_{i} + \sum_{j} n_{. j} lo g β_{j} .

Maximizing the two simplex problems independently gives

α_{i} = n_{i .} / N

and

β_{j} = n_{. j} / N

. Hence

θ_{ij}^{0} = \frac{n _{i .} n _{. j}}{N ^{2}} .

The fitted expected cell counts under independence are consequently

N θ_{ij}^{0} = n_{i .} n_{. j} / N

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