I know there are concentration bounds on $W(\mu,\hat{\mu})$ where $\mu$ and $\hat{\mu}$ are true and empirical distributions respectively, but is there anything out there on $W(\mu,\nu)$ versus $W(\hat{\mu},\hat{\nu})$ where $\hat{\mu}$ and $\hat{\nu}$ are the empirical distributions corresponding to $\mu$ and $\nu$ respectively?