By ”statistical prediction problem” the author means the following. Let X,Y be two random variables and \(P_{\theta}\) a family of probability distributions. A ”randomized prediction region” for Y given X is the set \(\{\) y: \(\phi\) (x,y)\(\geq u\}\), where \(\phi\) is a measurable function with values in [0,1] and u is a random variable uniformly distributed over [0,1] and independent of X and Y. The prediction region is of size 1- \(\epsilon\) if \(E_{\theta}\phi(X,Y)\geq 1-\epsilon\). The problem is to minimize \(E_{\theta}\{\int \phi(x,y)\xi(dx)\}\) amongst all prediction regions of size 1-\(\epsilon\). If the problem is in a sense invariant under the action of locally compact group, then the solution exists and is given by an explicit formula.
The following example is considered: Let \(X_ 1,...,X_ n,X_{n+1}\) be i.i.d. random vectors with probability density \(| \Lambda |^{- 1}f(\| \Lambda^{-1}(x-\mu)\|)\), where f is a known function and (\(\Lambda\),\(\mu)\) are unknown parameters. The problem is to predict the last \(p_ 2\) components of \(X_{n+1}\) given \(X_ 1,...,X_ n\) and the first \(p_ 1\) components of \(X_{n+1}\). An explicit formula for the best Euclidean invariant prediction region then follows.