Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 6H / c

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