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I am trying to find an unbiased estimator for (what looks like) the $L^2$ Wasserstein distance between two probability measures. I'm pretty sure that by bickel-lehmann, there is an unbiased estimator.

In order to find it, I am thinking of splitting the support of $F_0$ into a countable number of buckets (e.g. $[a,b]$ gets split into n equal parts, $\mathbb{R}$ gets split into $[n,n+1)$ etc.) and then you could estimate the integral as $\sum_{buckets}[(\sum_{i=1}^n\frac{\mathbb{1}_{X_i \in bucket}}{n}) - (F_0(sup_{bucket})-F_0(inf_{bucket})]^2$, as $(F_0(sup_{bucket})-F_0(inf_{bucket}))$ is the probability of laying within the bucket. Are there any obvious problems with this? I'm having problems showing its unbiased so I do doubt this estimator.

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  • $\begingroup$ I don't believe this is the $L_2$ Wasserstein distance, which would be $\{\int_0^1 (F^{-1} - F_0^{-1})^2 \ \mathrm d x\}^{1/2}$ $\endgroup$ – guy Oct 2 '18 at 3:52
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The limit of integral in real numbers, I supose. Why not using numerical integration? I can not read the formula you have written which involves summation. Can i ask you about the reference where you have copied the question and the answer which you have proposed?

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  • $\begingroup$ This is not an answer to the question. $\endgroup$ – Michael R. Chernick Jul 23 at 0:47

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