Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 6 / i / Solution

The parameter range is $0\le \theta \le 1$. Omitting data-only constants from the multinomial distribution likelihood, its log isFor an interior maximizer, the likelihood score must vanish:Multiply by $\theta (1-\theta )(7+4\theta )$ and collect powers to obtainThe likelihood is concave: its second derivative in the interior isTherefore an admissible interior root is the unique maximum. Endpoints must also be handled if some counts vanish. If $x_3>0$ and $x_1+x_2>0$, both endpoints have zero likelihood and the root is interior. If the maximum is at zero then $x_3=0$, and zero also satisfies the displayed polynomial; a maximum at one requires $x_1+x_2=0$, and one likewise satisfies it. Multiplying the score can introduce other endpoint roots, so not every polynomial root is an MLE: evaluate the actual likelihood on the feasible candidates.