Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 27J (Principles of Statistics)

A statistic is sufficient when the conditional distribution of the observation given it does not depend on the parameter. It is minimal sufficient when it is, up to null sets, a measurable function of every sufficient statistic. For a dominated family, the likelihood-ratio criterion identifies two observations precisely when their likelihood ratio is independent of the parameter.

The Rao–Blackwell theorem says that conditioning a square-integrable unbiased estimator on a sufficient statistic preserves unbiasedness and cannot increase variance. The Cramér–Rao bound for an unbiased estimator of scalar $\theta$ is ${\mathrm{Var}}_{\theta }\hat{\theta }\ge 1/I(\theta )$. It requires parameter-independent support, differentiability, interchange of differentiation and integration, and finite positive Fisher information. More generally the numerator is the square of the derivative of the estimand. The bound follows from $E_{\theta }[\hat{\theta }{\partial }_{\theta }\log f]=1$ and Cauchy–Schwarz.

For the uniform sample, the joint likelihood is ${\theta }^{-n}{1}_{\{0<x_i<\theta \forall i\}}$. Its dependence on the observations is through $T={\max }_i{X}_i$, proving sufficiency. Two such likelihoods have parameter-independent ratio exactly when their maxima coincide, proving minimality. The density of $T$ is $n{t}^{n-1}/{\theta }^n$ on $(0,\theta )$. Unbiasedness of $h(T)$ saysDifferentiating this absolutely continuous identity gives $h(t)=(n+1)t/n$ almost everywhere. Conversely that function is unbiased. Every finite-variance unbiased estimator, after Rao–Blackwellization, must therefore give this same function of $T$. Its variance cannot exceed that of any such estimator. ThusThe Cramér–Rao argument does not apply because the support depends on $\theta$; in particular differentiation of the support boundary invalidates the regular score identity.