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Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small perturbations to $f$. Are there any statistical metrics $D(\cdot,\cdot)$ (e.g. Kolmogorov-Smirnov, Wasserstein, Prokhorov, etc.) for which we can say something about the derivative $$\frac{d}{d\epsilon} D(f,f_1)$$ or $$\frac{d}{d\epsilon} D(f,f_2)$$ evaluated at $\epsilon=0$? How about if $f$ were, say, an exponential distribution instead?

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According to Proposition 7 on p. 236 of Givens and Shortt, the $L^2$ Wasserstein distance between $N(\mu_1,\sigma_1^2)$ and $N(\mu_2,\sigma_2^2)$ is the Euclidean distance between the points $(\mu_1,\sigma_1)$ and $(\mu_2,\sigma_2)$. So, the derivative of this Wasserstein distance is straightforward.

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