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? Commented Sep 30, 2015 at 17:56
• Since residuals can be grouped in a random vector, I'd like to know its distribution. At least first two moments. Commented Sep 30, 2015 at 18:35
• Got it, thanks! Have you considered simulating and plotting? Commented Sep 30, 2015 at 18:39
• Yes, I want to construct a confidence region for residuals. For instance, for Gaussian errors the region is an ellipsoid. Commented Sep 30, 2015 at 19:07

The residuals (actually called errors) are assumed to be randomly distributed with a double-exponential distribution (Laplace distribution). If you are fitting this x and y data points, do it numerically. You first calculate beta-hat_ML for these points as a whole using the formula you posted above. This will determine a line through the points. Then subtract each point's y value from the y value of the line at that x value. This is is the residual for that point. The residuals of all points can be used to construct a histogram that will give you the distribution of the residuals.

There is a good mathematical article on it by Yang (2014).

--Lee

• The link does not work. Commented Jun 10, 2017 at 16:12

I think this is equivalent to Robust Regression. In Robust Regression you minimize the 1-norm, instead of the 2-norm - and try to find $${\arg\min }_{\boldsymbol \beta \in \mathbb R^m} \sum _{i=1}^n |\mathbf x_i \cdot \boldsymbol \beta - y_i|$$ as you wrote.

One way to solve it is to approximate the 1-norm with a surrogate, like the Huber loss: i.e. $$h_\eta(x)=\sqrt {x^2+\eta}$$ for some small smoothing parameter $$\eta$$. So now the loss is $$\sum _{i=1}^n \sqrt{(\mathbf x_i \cdot \boldsymbol \beta - y_i)^2+\eta}$$ and you can use something like Gradient-Descent on this (now differentiable) function.

Here's some code I wrote for an HW exercise in MATLAB:

fun_g = @(u) sum( sqrt (u.^ 2 + eta^ 2 ));
fun_f = @(w) fun_g(X*w-z);
grad_g = @(u) u.*( 1. /( sqrt (u.^ 2 + eta^ 2 )));