What is a sufficient statistic of this distribution?

Question

Let $f_x(x)=\frac{e^{-|x-\theta|}}{2}$ be the distribution for x. Can you find a sufficient statistic for $\theta$?

I've obtained a likelihood function $L(\theta|\underline{x})=\frac{1}{2}^ne^{-\sum|x_i-\theta|}$, but $\sum|x_i-\theta|$ can't be a sufficient statistic for $\theta$ since I need information about $\theta$.

Answer 1

There is no sufficient statistic besides the trivial $T = (X_1,X_2,\dots,X_n)$. Take a look at the Fisher-Neyman factorization theorem.