When if ever is a median statistic a sufficient statistic?

I came across a remark on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic but, besides the obvious case of one or two observations where it equals the sample mean, I cannot think of another non-trivial and iid case where the sample median is sufficient.

• Did you mean to write "that a sample median could often be"? – Juho Kokkala Nov 6 '14 at 13:55
• It's an interesting question; the double exponential has the median for a ML estimator of its location parameter, but it's not sufficient. – Glen_b Nov 6 '14 at 15:57

In the case when the support of the distribution does not depend on the unknown parameter θ, we can invoke the (Fréchet-Darmois-)Pitman-Koopman theorem, namely that the density of the observations is necessarily of the exponential family form, $$\exp\{ \theta T(x) - \psi(\theta) \}h(x)$$ to conclude that, since the natural sufficient statistic $$S=\sum_{i=1}^n T(x_i)$$ is also minimal sufficient, then the median should be a function of $S$, which is impossible: modifying an extreme in the observations $x_1,\ldots,x_n$, $n>2$, modifies $S$ but does not modify the median.
In the alternative case when the support of the distribution does depend on the unknown parameter θ, we can consider the case when $$f(x|\theta) = h(x) \mathbb{I}_{A_\theta}(x) \tau(\theta)$$ where the set $A_\theta$ indexed by θ is the support of $f$. In that case, the factorisation theorem implies that $$\prod_{i=1}^n \mathbb{I}_{A_\theta}(x_i)$$ is a 0-1 function of the sample median $$\prod_{i=1}^n \mathbb{I}_{A_\theta}(x_i) = \mathbb{I}_{B^n_\theta}(\text{med}(x_{1:n}))$$ Adding a further observation $x_{n+1}$ which value is such that it does not modify the sample median then leads to a contradiction since it may be in or outside the support set, while $$\mathbb{I}_{B^{n+1}_\theta}(\text{med}(x_{1:n+1}))=\mathbb{I}_{B^n_\theta}(\text{med}(x_{1:n}))\times \mathbb{I}_{A_\theta}(x_{n+1})$$
• What is set $B_\theta^n$? – 3x89g2 May 3 '17 at 5:56