Moment-Matching and Best Entropy Estimation

Given the first n moments of an unknown function x̄ on the unit interval, a common estimate of x̄ is ψ(πn), where πn is a polynomial of degree n taking values in a prescribed interval, ψ is a given monotone function, and πn is chosen so that the moments of ψ(πn) equal those of x̄. This moment-matching procedure is closely related to best entropy estimation of x̄: two classical cases arise when ψ is the exponential function (corresponding to the Boltzmann-Shannon entropy) and the reciprocal function (corresponding to the Burg entropy). General conditions ensuring the existence and uniqueness of πn are given using convex programming duality techniques, and it is shown that the estimate ψ(πn) converges uniformly to x̄ providing x̄ is sufficiently smooth.