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

Abstract

We consider estimation of the parameter B in a multivariate linear functional relationship X i =ξ i+ξ 1i , Y i =Bξ i+ξ 2i , i=1,…,n , where the errors (ζ 1 i ′, ζ 2 i ′) are independent standard normal and ( ξ i , i ∈ N ) 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.