MINIMIZING FISHER INFORMATION OVER MIXTURES OF DISTRIBUTIONS

Abstract

SUMMARY. We study the problem of minimizing Fisher information over the class of distributions of the form J F(. , y) v(dy), where F(. , y), ? oo<y<oo, is a parametric family of distributions and v is an arbitrary probability measure on E c[ ?oo, oo]. We show, under suit able regularity conditions on F(., y), that the v minimizing Fisher information concentrates all its mass on a countable subset of isolated points in ( ? oo, oo) plus possibly { ? oo}. Applications are given to problems arising in robust estimation, robust smoothing, and minimax MSE estima tion of the mean of a normal distribution under various constraints.