An asymptotic minimax risk bound for estimation of a linear functional relationship

We consider estimation of the parameter B in a multivariate linear functional relationship Xi=[xi]i+[xi]1i, Yi=B[xi]i+[xi]2i, i=1,...,n, where the errors ([zeta]1i', [zeta]2i') are independent standard normal and ([xi]i, i [set membership, variant] ) is a sequence of unknown nonrandom vectors (incidental parameters). If there are no substantial a priori restrictions on the infinite sequence of incidental parameters then asymptotically the model is nonparametric but does not fit into common settings presupposing a parameter from a metric function space. A special result of the local asymptotic minimax type for the m.1.e. of B is proved. The accuracy of the normal approximation for the m.l.e. of order n-1/2 is also established.