Suppose that we have a parametric distribution $P_{\theta}$, which is indexed by the parameter $\theta \in \mathbb{R}^d$. Let $W\{\cdot,\cdot\}$ be the Wasserstein Metric between two distributions.
What is the second-order expansion of $W\{P_{\theta+\Delta \theta}, P_{\theta}\}$ in terms of $\Delta \theta$ and the density of $P_{\theta}$? Moreover, what about the expansion of more general integral probability metrics (IPMs), for example, Dudley metric and Maximum Mean Discrepancy?
