New Weak Error bounds and expansions for Optimal Quantization

Vincent Lemaire (), Thibaut Montes and Gilles Pagès ()
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Vincent Lemaire: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique
Thibaut Montes: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique, ICA - The Independent Calculation Agent
Gilles Pagès: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique

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Abstract: We propose new weak error bounds and expansion in dimension one for optimal quantization-based cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewise-defined locally Lipschitz or α-Hölder derivatives. This new results rest on the local behaviors of optimal quantizers, the L r-L s distribution mismatch problem and Zador's Theorem. This new expansion supports the definition of a Richardson-Romberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.

Keywords: Optimal quantization; Numerical integration; Weak error; Romberg extrapolation; Variance reduction; Monte Carlo simulation; Product quantizer; Product quantizer 2010 AMS Classification: 65C05; 60E99; 65C50 (search for similar items in EconPapers)
Date: 2020
Note: View the original document on HAL open archive server: https://hal.science/hal-02361644v3
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Citations: View citations in EconPapers (3)

Published in Journal of Computational and Applied Mathematics, inPress, 371, pp.112670. ⟨10.1016/j.cam.2019.112670⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-02361644

DOI: 10.1016/j.cam.2019.112670

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