Is the median a sufficient statistic for a uniform distribution on $(-θ, θ)$?

Question

I have a uniform distribution on $(-θ, θ)$ and I have to find a sufficient statistic. I know that the order statistic [$x_{(1)}$, $x_{(n)}$] are jointly minimal sufficient but I was wondering whether I can say that [$x_{(1)}$, $x_{(m)}$, $x_{(n)}$] is sufficient for $θ$. (where $x_{(m)}$ is the median such that if I have 10 observation $x_{(m)}$ would be $x_{(5)}$.)

Answer 1

Sufficiency of the order statistic implies (in this case) that the mapping you have written out is also sufficient, by the Fisher-Neyman factorization lemma. Intuitively, a sufficient statistic compresses the data to only contain the parts that are 'relevant' for estimating $\theta$. Since you have added no new information on $\theta$, but also not removed the minimum and maximum the statistic will still be sufficient. You can show this pretty easily: Use that the order statistic is sufficient. Then you can factorize the density into two functions, one of $X$ and one of $(\theta, T(X))$ (where $T$ is the statistic). Construct $\tilde{T}$ by $$ \tilde{T} = T(X) + 0\cdot X_{(m)} $$ Then you can surely plug in $\tilde{T}$ instead of $T$, and then you will have still factorized the density. As such, the statistic is sufficient.

It is worth it, however, to note that the median statistic in itself is not sufficient for $\theta$, however.