Minimax rate for optimal transport regression between distributions. (English) Zbl 1502.62043
Summary: 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.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62G05 | Nonparametric estimation |
| 62J05 | Linear regression; mixed models |
| 62G20 | Asymptotic properties of nonparametric inference |
References:
| [1] | Aneiros, G.; Cao, R.; Fraiman, R.; Genest, C.; Vieu, P., Recent advances in functional data analysis and high-dimensional statistics, J. Multivariate Anal., 170, 3-9 (2019) · Zbl 1415.62043 |
| [2] | Brunel, É.; Mas, A.; Roche, A., Non-asymptotic adaptive prediction in functional linear models, J. Multivariate Anal., 143, 208-232 (2016) · Zbl 1328.62408 |
| [3] | Chagny, G.; Roche, A., Adaptive and minimax estimation of the cumulative distribution function given a functional covariate, Electron. J. Stat., 8, 2, 2352-2404 (2014) · Zbl 1302.62082 |
| [4] | Chen, Y.; Lin, Z.; Müller, H.-G., Wasserstein regression, J. Amer. Statist. Assoc., 1-14 (2021) |
| [5] | Cuevas, A., A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference, 147, 1-23 (2014) · Zbl 1278.62012 |
| [6] | Ghodrati, L.; Panaretos, V. M., Distribution-on-distribution regression via optimal transport maps, Biometrika, 109, 957-974 (2022) · Zbl 07638095 |
| [7] | Goia, A.; Vieu, P., An introduction to recent advances in high/infinite dimensional statistics, J. Multivariate Anal., 146, 1-6 (2016) · Zbl 1384.00073 |
| [8] | Hall, P.; Horowitz, J. L., Methodology and convergence rates for functional linear regression, Ann. Statist., 35, 1, 70-91 (2007) · Zbl 1114.62048 |
| [9] | Hsing, T.; Eubank, R., Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators (2015), John Wiley & Sons · Zbl 1338.62009 |
| [10] | Ling, N.; Vieu, P., Nonparametric modelling for functional data: selected survey and tracks for future, Statistics, 52, 4, 934-949 (2018) · Zbl 1411.62084 |
| [11] | Morris, J. S., Functional regression, Annu. Rev. Stat. Appl., 2, 321-359 (2015) |
| [12] | Panaretos, V. M.; Zemel, Y., Statistical aspects of wasserstein distances, Annu. Rev. Stat. Appl., 6, 405-431 (2019) |
| [13] | Panaretos, V. M.; Zemel, Y., An Invitation to Statistics in Wasserstein Space (2020), Springer Nature · Zbl 1433.62010 |
| [14] | Petersen, A.; Zhang, C.; Kokoszka, P., Modeling probability density functions as data objects, Econom. Stat., 21, 159-178 (2022) |
| [15] | Van Der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes (1996), Springer · Zbl 0862.60002 |
| [16] | Wainwright, M. J., High-Dimensional Statistics: A Non-Asymptotic Viewpoint (2019), Cambridge University Press · Zbl 1457.62011 |
| [17] | Zhang, C.; Kokoszka, P.; Petersen, A., Wasserstein autoregressive models for density time series, J. Time Series Anal., 43, 1, 30-52 (2022) · Zbl 1493.62182 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
