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Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs

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  • Tengyuan Liang

    (University of Chicago - Booth School of Business)

Abstract

We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on n independent samples from them. Curiously, we show that estimating the IPM itself between probability measures is not significantly easier than estimating the probability measures under the IPM. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a log log(n)/ log(n) factor.

Suggested Citation

  • Tengyuan Liang, 2020. "Estimating Certain Integral Probability Metrics (IPMs) Is as Hard as Estimating under the IPMs," Working Papers 2020-153, Becker Friedman Institute for Research In Economics.
    • Handle: RePEc:bfi:wpaper:2020-153
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    File URL: https://repec.bfi.uchicago.edu/RePEc/pdfs/BFI_WP_2020153.pdf
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    References listed on IDEAS

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    1. Max Sommerfeld & Axel Munk, 2018. "Inference for empirical Wasserstein distances on finite spaces," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(1), pages 219-238, January.
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