# consistency of mle of double exponential distribution ( not advanced)

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 $$y_i$$ is.

I prove that $$\hat\mu = med(y_i)$$ is consistent.

How can I show that the $$\hat\sigma$$ is consistent

I want only use WLLN or definition of convergence in probability or properties of distribution.

• Use the results at stats.stackexchange.com/questions/45124 along with a definition of a consistent estimator. It might help to forget even that the distribution is double exponential: it suffices that it has a definite median and the probability doesn't vanish too quickly in small neighborhoods of that median.
– whuber
Nov 22, 2021 at 23:16