Estimating the Wasserstein-1 metric between two beta-Holder smooth densities has minimax error n^{-(beta+1)/(2beta+d)} up to a log log n / log n factor.
Density estimation by wavelet thresholding
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On the Minimax Optimality of Estimating the Wasserstein Metric
Estimating the Wasserstein-1 metric between two beta-Holder smooth densities has minimax error n^{-(beta+1)/(2beta+d)} up to a log log n / log n factor.