Hausdorff moment problem: Recovery of an unknown support for a probability density function

Abstract

Abstract In this work the following problem is discussed: Unknown are two objects, a probability density (corresponding to a distribution, or to a probability measure) f and its support supp ⁡ { f } = [ L , U ] {\operatorname{supp}\{f\}=[L,U]} which is assumed to be bounded/compact. Available is the sequence of all integer order moments of f and the goal is to recover the support [ L , U ] {[L,U]} or both the support [ L , U ] {[L,U]} and the density f. This paper provides the sequences involving integer order moments that approximate the compact support of a probability measure. The proposed sequences are based on the ratios of subsequent moments and they converge faster than the previously suggested sequences representing the n-th roots of n-th moments. As a result, the approximation of a probability density function with unknown support is proposed. The convergence of newly defined approximations of the boundaries and the densities are demonstrated through various examples, including the case of noisy data.