Question: Let \(y_{1}, \ldots, y_{n}\) be a random sample from \(f(y ; \mu, \sigma)=(2 \sigma)^{-1} \exp (-|y-\mu| / \sigma),-\infty

\(\sigma>0 ;\) this is the Laplace density.

1. Write down the log likelihood for \(\mu\) and \(\sigma\) and by showing that \(^{1}\)
   \[\frac{d}{d \mu} \sum\left|y_{j}-\mu\right|=\#\left\{y_{j}<\mu\right\}-\#\left\{y_{j}>\mu\right\}=n-2 R,\]
   where \(R=\#\left\{y_{j}>\mu\right\}\), show that for any fixed \(\sigma>0\) the maximum likelihood estimate of \(\mu\) is \(\widehat{\mu}=\operatorname{median}\left\{y_{j}\right\}\), and deduce that the maximum likelihood estimate of \(\sigma\) is the mean absolute deviation \(\widehat{\sigma}=n^{-1} \sum\left|y_{i}-\widehat{\mu}\right|\)
2. Show that in large samples \(\widehat{\mu} \sim N\left(\mu, \sigma^{2} / n\right)\) and \(\widehat{\sigma} \stackrel{P}{\longrightarrow} \sigma\).
3. Is this a regular model for maximum likelihood estimation?

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