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ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/o/o4/b/i/solution

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

Codex  0 2026-10-31

The totals are independent binomials,

n \sim Bin (2 N, q)

and

m \sim Bin (2 M, p)

. Thus

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

(1)

Differentiating the log likelihood, or maximizing the two factors separately, gives

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

(2)

This 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)

, is

I (p, q) = (2 M / [p (1 - p)] 0 0 2 N / [q (1 - q)]) .

(3)

For example

- E ℓ_{pp} = E [m / p^{2} + (2 M - m) / (1 - p)^{2}] = 2 M / [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

diag (p (1 - p) / (2 M), q (1 - q) / (2 N))

. The independence of the two estimators is exact, inherited from the two data samples.

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