# A sufficient statistic for Laplace distribution

Suppose we have p dimensional vector of $X =[X_1 \dots X_n]$ where X is Laplace distributed. What will be a sufficient statistics for estimating covariance of $X$?

Would it be the sample covariance, i.e.$$\frac{1}{N}\sum_{i=1}^{N} (X_i-\bar X)(X_i-\bar X)^T\,?$$

• Can you give us the density of the multivariate Laplace distribution? Jul 29, 2015 at 12:36
• The point of the comment by @Xi'an is that there is more than one meaning to a (multivariate) Laplace distribution: see stats.stackexchange.com/questions/52276/….
– whuber
Jul 29, 2015 at 12:37
• If we use the density given in the tagged link I get sufficient statistic as $\frac{1}{N}\sqrt{(X-\mu)(X-\mu)^T}^{2-p/2}$ but I am not sure if it is correct. Can we take sample covariance itself as sufficient statistic assuming one to one? Jul 29, 2015 at 13:00
• You did not read properly the definition of this distribution: the correct answer is that there is no non-trivial sufficient statistic. This is because the Laplace distribution is not an exponential family of distributions. Jul 29, 2015 at 13:02

Ìf by multivariate Laplace distribution you mean any distribution such that the marginals all are distributed from unidimensional Laplace distributions, with densities$$f(x_i|\mu_i,\sigma_1)=\dfrac{1}{2\sigma_i}\exp\left\{-|x_i-\mu_i|/\sigma_i\right\},$$then there cannot be a sufficient statistic of fixed dimension for the parameters $(\mathbf{\mu},\mathbf{\Sigma})$ of the joint distribution. This is because the distribution cannot belong to an exponential family, hence cannot have a sufficient statistic of fixed dimension by virtue of the (Darmois-)Pitman-Koopman lemma.
• +1 It does leave one wondering, though, what a sufficient statistic for any specific $n$ would be.