MLE of $\beta$ and $\sigma^2$ in linear regression with Laplace errors

Question

Say we have a linear regression model $$ y_i = x_i^T\beta + \sigma\varepsilon_i $$ for $i = 1, \dots, n$, and $\varepsilon_i$ is distributed according to a Laplace distribution $\mathcal{L}(0, 1)$ with density $f(\varepsilon) = 2^{-1}e^{-|\varepsilon|}$, then what are the MLE estimators for $\beta$ and $\sigma^2$?

So far for the $\beta_{MLE}$, I have gotten down to $\beta_{MLE} = \underset{\beta}{\operatorname{argmin}} \sum_i |y_i - x_i^T\beta|$ but now I do not know how to solve it. I attempted to differentiate wrt $\beta$ and then set that to 0, where I got $\sum_i -x_i\operatorname{sign}{(y_i-x_i^T\beta)} = 0$. I do not know how to proceed from here to find $\beta_{MLE}$.

For the $\sigma^2_{MLE}$, I have reduced it down to $\sigma^2_{MLE} = \underset{\sigma^2}{\operatorname{argmin}} \sum_i \sigma^{-1}|y_i-x_i^T\beta|$. I attempted to differentiate wrt $\sigma$ and then set that to 0, where I got $-\sigma^{-2}\sum_i |y_i-x_i^T\beta| = 0$. I do not know how to proceed from here to find $\sigma^2_{MLE}$.

Guidance would be much appreciated!

Answer 1

Like Gaussian errors are equivalent to the least squares method and estimating the mean; This problem is equivalent to quantile regression and estimating the median. There are several approaches to solve it.

The scale of the errors, $\sigma$, can be found seperately by using the method of moments, or by maximizing the entire log likelihood function (after filling in the solution for the optimum of $\beta$, which can be found seperately from the optimum for $\sigma$)

$$\mathcal{L}(\sigma,\boldsymbol{\beta}) = - n \log(2\sigma) -\frac{1}{\sigma}\sum_{i=1}^n |X_i\boldsymbol{\beta} - y_i|$$

Somewhere along the way of your derivations you have lost this $ - n \log(2\sigma)$ term.