 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/o/o4/b/solution

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

Maximizing the two terms over their respective probability simplices, by the same Lagrange multiplier argument as in part (a), gives

α_{i} = \frac{n _{i .}}{N}, β_{j} = \frac{n _{. j}}{N} .

Therefore the constrained maximum-likelihood estimates are

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

The associated fitted cell counts are

E_{ij} = N θ_{ij}^{0} = n_{i .} n_{. j} / N

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