Hausdorff moment problem and Maximum Entropy: On the existence conditions

doi:10.1016/j.amc.2011.05.082

Applied Mathematics and Computation

Volume 218, Issue 2, 15 September 2011, Pages 430-433

https://doi.org/10.1016/j.amc.2011.05.082 Get rights and content

Abstract

Different existence conditions of the Maximum Entropy solution to finite Hausdorff moment problem have been formulated in literature. Through a counterexample we prove that the most cited one is uncorrect. We do not bound ourselves to a crude counterexample, as we think that a detailed explanation is of interest by itself. It clarifies the difference existing between the finite and infinite Hausdorff moment problem existence conditions.

Section snippets

Hausdorff moment problem and Maximum Entropy

The finite Hausdorff moment problem consists of recovering an unknown probability density function (pdf) f(x), with support D = [0, 1], from the knowledge of its associated sequence ${μ_{j}}_{j = 0}^{M}$ of integer moments, with $μ_{j} = \int_{0}^{1} x^{j} f (x) dx, j ⩾ 0, μ_{0} = 1$ . In actual practice, the problem consists of determining approximations to the unknown underlying function f(x) from a finite collection ${μ_{j}}_{j = 1}^{M}$ of its integer moments. Then we settle for a reasonable analytical form of f(x), and the Maximum Entropy (MaxEnt)

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There are more references available in the full text version of this article.

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It is known that the PDF can be uniquely determined if a infinite number of statistical moments are provided, which are always unavailable in practices. Instead, a reduced number of statistical moments are always used for the reconstruction of the PDF, which leads to the so-called classical moment problem [1–3]. That is, the reconstruction of a PDF using a finite number of its statistical moments could be an ill-posed inverse problem, where the existence and the uniqueness of the PDF may be questioned.
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Hausdorff moment problem and Maximum Entropy: On the existence conditions

Abstract

Section snippets

Hausdorff moment problem and Maximum Entropy

Appl. Math. Comput.

Maximum entropy in the problem of moments

J. Math. Phys.

The Markov Moment Problem and Extremal Problems