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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.

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Nov 22, 2021 at 23:16

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