New weak error bounds and expansions for optimal quantization

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 behaviours 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.

Representative Points Based on Power Exponential Kernel Discrepancy

“…[1,5] extended the rep-points for the elliptical distributions. [6,7] used the rep-points as the refined Monte Carlo technique for approximating the integration or expectation. More applications of rep-points can be found in the uncertainty quantification [2,8,9] and Bayesian analysis [10][11][12].…”

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“…For more results on the rate of convergence, the value of α, we refer to [Pag18] for a survey in R d and to [LMP19] for recent improved results in the one-dimensional case.…”