Past exam of the options II course of the University of Oxford / 2010 / O / O4 / b / i

Solution

The two totals are independent, with

n \sim Bin (2 N, q)

and

m \sim Bin (2 M, p)

. Their joint likelihood is

L (p, q) = (n 2 N) q^{n} (1 - q)^{2 N - n} (m 2 M) p^{m} (1 - p)^{2 M - m} .

Differentiating the separated log-likelihood gives

p = \frac{m}{2 M}, q = \frac{n}{2 N} .

These formulas include all-zero or all-one totals as boundary maxima. For interior

p, q

(p, q)

I (p, q) = \frac{2 M}{p ( 1 - p )} 0 0 \frac{2 N}{q ( 1 - q )} .

The off-diagonal entries vanish because the likelihood separates, and the two estimates are independent even at finite sample size. For large

M, N

, their joint distribution is approximately

(p q) \sim N ((p q), (p (1 - p) / (2 M) 0 0 q (1 - q) / (2 N))) .

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