Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis

Ken'ichiro Tanaka and Alexis Akira Toda

University of California at San Diego, Economics Working Paper Series from Department of Economics, UC San Diego

Abstract: The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating distribution by minimizing the Kullback-Leibler information (relative entropy) of the unknown discrete distribution relative to an initial discretization based on a quadrature formula subject to some moment constraints. We study the theoretical error bound and the convergence of this approximation method as the number of discrete points increases. We prove that (i) the theoretical error bound of the approximate expectation of any bounded continuous function has at most the same order as the quadrature formula we start with, and (ii) the approximate discrete distribution weakly converges to the given continuous distribution. Moreover, we present some numerical examples that show the advantage of the method and apply to numerically solving an optimal portfolio problem.

Keywords: probability distribution; discrete approximation; generalized moment; quadrature formula; Kullback-Leibler information; Fenchel duality; Pure Mathematics; Applied Mathematics; Numerical and Computational Mathematics; Numerical & Computational Mathematics (search for similar items in EconPapers)
Date: 2015-01-01
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Citations: View citations in EconPapers (9)

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Working Paper: Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis (2015)
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Persistent link: https://EconPapers.repec.org/RePEc:cdl:ucsdec:qt2tc0m67t

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