Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / O / O4 / b / i / Solution

The totals are independent binomials, $n\sim \mathrm{Bin}(2N,q)$ and $m\sim \mathrm{Bin}(2M,p)$. ThusDifferentiating the log likelihood, or maximizing the two factors separately, givesThis remains the constrained maximum at zero or one when a total is at an endpoint. At interior parameters the sample information, in coordinate order $(p,q)$, isFor example $-\mathbb{E}{\ell }_{pp}=\mathbb{E}[m/p^2+(2M-m)/(1-p{)}^2]=2M/[p(1-p)]$; the cross derivative is zero. As both sample sizes grow with nondegenerate interior probabilities, the joint approximation is normal with mean $(p,q)$ and covariance $\mathrm{diag}(p(1-p)/(2M),q(1-q)/(2N))$. The independence of the two estimators is exact, inherited from the two data samples.