Motivated by this question, assume we have independent samples $(X_i)_{i=1}^{\infty}$ from a Cauchy distribution with unknown median $a\in\mathbb{R}$ and scale parameter $b$. What is the best way to estimate $a$?
More precisely, for $d$ some distance between probability distributions of your choice what choice of $f\colon \mathbb{R}^{n}\to\mathbb{R}$ minimizes
$$ \sup_{a\in\mathbb{R}}d(\mu_{a,f},\delta_{a}), $$ where $\mu_{a,f}$ is the distribution of $f(X_1,\dots,X_n)$ when $X_i$ have median $a$, and how does the minimum behave as $n\to\infty$?
For example, if we use $f(x_1,\dots,x_n):=\frac{1}{n}\sum_{i=1}^{n}x_i$, then $f(X_1,\dots,X_n)$ has the same distribution as $X_1$ and thus any distance between $\mu_{a,f}$ and $\delta_{a}$ will be independent of $n$ and in particular not converge to zero.