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Year 2 of the mathematics course of the University of Oxford

Options II course of the University of Oxford

Past exam of the options II course of the University of Oxford

2010

The two totals are independent, with $n\sim\operatorname{Bin}(2N,q)$ and $m\sim\operatorname{Bin}(2M,p)$. Their joint likelihood is
$$
L(p,q)=\binom{2N}nq^n(1-q)^{2N-n}
\binom{2M}mp^m(1-p)^{2M-m}.
$$
Differentiating the separated <log-likelihood> gives
$$
\boxed{\widehat p=\frac m{2M},\qquad\widehat q=\frac n{2N}.}
$$
These formulas include all-zero or all-one totals as boundary maxima. For interior $p,q$, the <Fisher information> in the ordered parameter vector $(p,q)$ is
$$
\mathcal I(p,q)=\begin{pmatrix}
\dfrac{2M}{p(1-p)}&0\\[2mm]
0&\dfrac{2N}{q(1-q)}
\end{pmatrix}.
$$
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
$$
\boxed{\begin{pmatrix}\widehat p\\\widehat q\end{pmatrix}
\sim N\left(\begin{pmatrix}p\\q\end{pmatrix},
\begin{pmatrix}p(1-p)/(2M)&0\\0&q(1-q)/(2N)\end{pmatrix}\right).}
$$

Solution

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

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

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