Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 28L (Principles of Statistics)

For an observed value $X=x$, the likelihood iswhile the gamma prior with shape $\alpha$ and rate $\lambda$ has densityThe posterior density is therefore proportional toNormalizing this gamma kernel proves the Poisson-gamma conjugacy formula

The risk of a decision rule is its expected loss at a fixed parameter value:For a prior $\pi$, a Bayes estimator minimizes the integrated riskor equivalently minimizes posterior expected loss for almost every observation. A minimax decision rule minimizesover all decision rules.

Let ${\delta }^{\prime }$ be any decision rule. For every $j$,because $r_j$ is the infimum of the integrated risk under ${\pi }_j$. Taking the lower limit givesThe given rule $\delta$ has constant risk $r$, and hence worst-case risk exactly $r$. It attains this universal lower bound and is therefore minimax. This is the constant-risk limit-of-Bayes-risk criterion.

Given $X=x$, the posterior expected loss from reporting $d$ isThis is a strictly convex quadratic in $d$ whenever the conditional inverse moment is finite and positive. Differentiating with respect to $d$ givesso the Bayes estimator under reciprocal weighted quadratic loss is

For $\delta (X)=X$ and $X\sim \mathrm{Pois}(\theta )$,Thus this rule has constant risk one.

To show that no rule has smaller worst-case risk, fix any $\alpha >1$ and letPart (a) givesSince the reciprocal moment of a gamma variable with shape $k>1$ and rate $s$ is $s/(k-1)$, part (b)(iii) gives the Bayes ruleIts minimum posterior expected loss isHence the Bayes risk isThe criterion from part (b)(ii) now proves