Linear regression with Laplace errors

Consider a linear regression model: $$y_i = \mathbf x_i \cdot \boldsymbol \beta + \varepsilon _i, \, i=1,\ldots ,n,$$ where $\varepsilon _i \sim \mathcal L(0, b)$, that is, Laplace distribution with $0$ mean and $b$ scale parameter, are all are mutually independent. Consider a maximum likelihood estimation of unknown parameter $\boldsymbol \beta$: $$-\log p(\mathbf y \mid \mathbf X, \boldsymbol \beta, b) = n\log (2b) + \frac 1b\sum _{i=1}^n |\mathbf x_i \cdot \boldsymbol \beta - y_i|$$ from which $$\hat{\boldsymbol \beta}_{\mathrm {ML}} = {\arg\min }_{\boldsymbol \beta \in \mathbb R^m} \sum _{i=1}^n |\mathbf x_i \cdot \boldsymbol \beta - y_i|$$

How can one find a distribution of residuals $\mathbf y - \mathbf X\hat{\boldsymbol \beta}_{\mathrm {ML}}$ in this model?

• What do you mean by find a distribution of residuals? – jlimahaverford Sep 30 '15 at 17:56
• Since residuals can be grouped in a random vector, I'd like to know its distribution. At least first two moments. – nmerci Sep 30 '15 at 18:35
• Got it, thanks! Have you considered simulating and plotting? – jlimahaverford Sep 30 '15 at 18:39
• Yes, I want to construct a confidence region for residuals. For instance, for Gaussian errors the region is an ellipsoid. – nmerci Sep 30 '15 at 19:07