Sufficiency of Sample Mean for Laplace Distribution I recently started reading about sufficient statistics. I have the following questions:
1) Is sample mean a sufficient statistic for Laplace Distribution (aka Double Exponential) if we already know the scale parameter?
2) If not, is this also related to the fact that it is not an efficient estimator of the mean/location? 
EDIT
For example sample median with known scale parameter is sufficient and efficient.
Thanks
 A: For on observation, the Laplace pdf is
$$f_X(x) = \dfrac 1 {2b} \exp(-\dfrac {|x-\mu|} b)$$
For multiple iid observations, the pdf is
$$f_\boldsymbol X(\boldsymbol x) = \dfrac 1 {(2b)^n} \exp(- \dfrac 1 b \sum_{i=1}^n {|x_i-\mu|})$$
The easiest way to determine what statistics are sufficient for $\boldsymbol X$ is to try to use the Factorization Theorem (https://en.wikipedia.org/wiki/Sufficient_statistic#Fisher.E2.80.93Neyman_factorization_theorem). However, if you start to work with this expression, you'll see that the absolute values in the sum make it impossible to do any simplification/factorization.
To answer your first question, the sample mean is not a sufficient statistic (event if $b$ is known). However, if $\mu$ is known, then $\sum_{i=1}^n |x_i-\mu|$ is a sufficient statistic for $b$. But $\mu$ will almost never be known unless it's assumed to be zero.
As for your second question, I don't believe there are any theorems which directly state for some conditions, inefficiency implies insufficiency or vice-versa. However, there are theorems which connect sufficient statistics to maximum likelihood estimators and MLEs are asymptotically efficient under certain regularity conditions. So in that sense, I suppose you could view the insufficiency and inefficiency of the sample mean as related results.
A: The sample median is the maximum likelihood estimator of the mean if the scale is known, but it is not the sufficient statistics. I agree that the order statistics is the sufficient statistics. This can be seen using the factorization theorem and examining the dependence of the RN derivative, upon the examination of the absolute values therein, on the data across all mu values.    
