MOMENT-MATCHING AND BEST ENTROPY ESTIMATION

Given the first n moments of an unknown function xBAR on the unit inte rval, a common estimate of xBAR is psi(pi(n)), where pi(n) is a polyno mial of degree n taking values in a prescribed interval, psi is a give n monotone function, and pi(n) is chosen so that the moments of psi(pi (n)) equal those of xBAR. This moment-matching procedure is closely re lated to best entropy estimation of xBAR: two classical cases arise wh en psi is the exponential function (corresponding to the Boltzmann-Sha nnon entropy) and the reciprocal function (corresponding to the Burg e ntropy). General conditions ensuring the existence and uniqueness of p i(n) are given using convex programming duality techniques, and it is shown that the estimate psi(pi(n)) converges uniformly to xBAR providi ng xBAR is sufficiently smooth. (C) 1994 Academic Press, Inc.