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Minimax rate for optimal transport regression between distributions

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  • Ghodrati, Laya
  • Panaretos, Victor M.

Abstract

Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower bound that matches the convergence rate attained by the Fréchet least squares estimator.

Suggested Citation

  • Ghodrati, Laya & Panaretos, Victor M., 2023. "Minimax rate for optimal transport regression between distributions," Statistics & Probability Letters, Elsevier, vol. 194(C).
    • Handle: RePEc:eee:stapro:v:194:y:2023:i:c:s0167715222002711
    DOI: 10.1016/j.spl.2022.109758
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    References listed on IDEAS

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    1. Brunel, Élodie & Mas, André & Roche, Angelina, 2016. "Non-asymptotic adaptive prediction in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 208-232.
    2. Chao Zhang & Piotr Kokoszka & Alexander Petersen, 2022. "Wasserstein autoregressive models for density time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 30-52, January.
    3. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    4. Laya Ghodrati & Victor M Panaretos, 2022. "Distribution-on-distribution regression via optimal transport maps [Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line]," Biometrika, Biometrika Trust, vol. 109(4), pages 957-974.
    5. Petersen, Alexander & Zhang, Chao & Kokoszka, Piotr, 2022. "Modeling Probability Density Functions as Data Objects," Econometrics and Statistics, Elsevier, vol. 21(C), pages 159-178.
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