Consider the linear regression setting where the responses $Y_{i}, i=1, \ldots, n$ are assumed independent with means $\mu_{i}=x_{i}^{\mathrm{T}} \beta$. Here $x_{i}$ is a vector of known explanatory variables and $\beta$ is a vector of unknown regression coefficients.

Show that if the response distribution is Laplace, i.e.,

$$Y_{i} \sim f\left(y_{i} ; \mu_{i}, \sigma\right)=(2 \sigma)^{-1} \exp \left\{-\frac{\left|y_{i}-\mu_{i}\right|}{\sigma}\right\}, \quad i=1, \ldots, n ; \quad y_{i}, \mu_{i} \in \mathbb{R} ; \sigma \in(0, \infty)$$

then the maximum likelihood estimate $\hat{\beta}$ of $\beta$ is obtained by minimising

$$S_{1}(\beta)=\sum_{i=1}^{n}\left|Y_{i}-x_{i}^{\mathrm{T}} \beta\right|$$

Obtain the maximum likelihood estimate for $\sigma$ in terms of $S_{1}(\hat{\beta})$.

Briefly comment on why the Laplace distribution cannot be written in exponential dispersion family form.