Let $y_i\sim DE(\mu, \sigma), i=1,2,...,n, \ i.i.d$.
Where $DE$ represents the double exponential distribution. The the MLE of $\sigma$ is:
$\hat\sigma$ = $\frac{1}{n} \sum_{i=1}^{n}|y_i-med(y_i)|$, where $med$ refers to the median of the $y_i$s.
How can I show that the MLE is consistent?
I have learned some methods to prove consistency of random variable, but I don't know how to deal with the absolute value and median.
To close this one, here is a way to prove consistency constructively, without invoking the general properties of the MLE that make it a consistent estimator.
Because the sample is i.i.d., it is ergodic-stationary. This means that sample moments and quantiles tend to their theoretical true counterparts. So
$$\hat \sigma =\frac{1}{n} \sum_{i=1}^{n}|y_i-med(y_i)| \to_p E|Y-\mu|$$
This is the probability limit of the estimator, and we have to show that $\sigma = E|Y-\mu|$ to prove consistency.
We have that $Y-\mu \equiv Z \sim DE(0,\sigma)$. So
$$E|Y-\mu|=E|Z| = \int_{-\infty}^{\infty}|z| \frac 1{2\sigma}\exp\{-|z|/\sigma\}dz$$
$$=2\int_{0}^{\infty}(z/\sigma) \frac 1{2}\exp\{-z/\sigma\}dz$$
Simplify $2$ and multiply and divide by $\sigma$
$$E|Y-\mu|=E|Z| = \sigma \int_{0}^{\infty}(z/\sigma) \exp\{-z/\sigma\}d(z/\sigma)$$
But the integrand is now the expected value of a standard exponential random variable, so the integral equals unity. Therefore we obtain
$$\text{plim } \hat \sigma = E|Y-\mu|= \sigma$$
