On a relation between higher order asymptotic risk sufficiency and higher order asymptotic sufficiency in a local sense (Q581952)

Let (\({\mathcal X},{\mathcal A}_ n,P_{\theta,n}),\theta \in \Theta\), be a family of probability spaces with \(\Theta\) being Euclidean. Let \(\{\) \({\mathcal B}_ n\}\) be a sequence of sub \(\sigma\)-fields of \(\{\) \({\mathcal A}_ n\}\), and \(r_ n^{{\mathcal B}_ n}(c;\theta,\theta ')\) be the Bayes risk of the problem of testing a hypothesis \(``P_{\theta ',n}\) is true'' against an alternative \(``P_{\theta,n}\) is true'' with an experiment (\({\mathcal X},{\mathcal B}_ n,\{P_{\theta ',n},P_{\theta,n}\})\) relative to a prior probability distribution \((c/(1+c)\), \(1/(1+c))\) on \(\{\theta ',\theta \}\). The author gives a sufficient condition for the Bayes risk \(r_ n^{{\mathcal B}_ n}\) for \(\{\) \({\mathcal B}_ n\}\) to be a higher order locally asymptotically sufficient sequence of \(\sigma\)- fields.