# Unbiased estimator for $L^2$ probability distance norm

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.

• I don't believe this is the $L_2$ Wasserstein distance, which would be $\{\int_0^1 (F^{-1} - F_0^{-1})^2 \ \mathrm d x\}^{1/2}$ – guy Oct 2 '18 at 3:52