Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 6H (Statistics)

A statistic $T(X)$ is sufficient for a parameter $\theta$ when the conditional distribution of the full sample $X$ given $T$ does not depend on $\theta$. For a dominated model, the Fisher-Neyman factorization theorem says this is equivalent to a factorizationwhere $h$ is independent of $\theta$.

A sufficient statistic $T$ is minimal sufficient if it is a function of every other sufficient statistic: whenever $U$ is sufficient, there is a function $h$ such that $T=h(U)$ almost surely, up to the usual null-set qualification. Thus its level sets give the coarsest sufficient partition of the sample space. In a dominated family with positive densities, the likelihood-ratio criterion for minimal sufficiency says that $T$ is minimal sufficient when

For $\sigma >0$, the density isso the Fisher-Neyman factorization theorem proves that $|X|$ is sufficient.

Moreover,is independent of $\sigma$ exactly when $x^2=y^2$, equivalently $|x|=|y|$. The likelihood-ratio criterion for minimal sufficiency therefore proves that

The joint density depends on the sample throughnot merely through ${\sum }_i|X_i|$. For example, after padding with zeros when $n>2$, takeBoth samples have sum of absolute values $2$, butwhich depends on $\sigma$. The likelihood-ratio necessary condition for sufficiency fails, so