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Laplace (Double Exponential, Bilateral Exponential) Distribution

$$\begin{eqnarray*} f\left(x\right) & = & \frac{1}{2}e^{-\left|x\right|}\\ F\left(x\right) & = & \left\{ \begin{array}{ccc} \frac{1}{2}e^{x} & & x\leq0\\ 1-\frac{1}{2}e^{-x} & & x>0\end{array}\right.\\ G\left(q\right) & = & \left\{ \begin{array}{ccc} \log\left(2q\right) & & q\leq\frac{1}{2}\\ -\log\left(2-2q\right) & & q>\frac{1}{2}\end{array}\right.\end{eqnarray*}$$

$$\begin{eqnarray*} m_{d}=m_{n}=\mu & = & 0\\ \mu_{2} & = & 2\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & 3\end{eqnarray*}$$

The ML estimator of the location parameter is

$$\hat{L}=\mathrm{median}\left(X_{i}\right)$$

where $X_{i}$ is a sequence of $N$ mutually independent Laplace RV’s and the median is some number between the $\frac{1}{2}N\mathrm{th}$ and the $(N/2+1)\mathrm{th}$ order statistic ( e.g. take the average of these two) when $N$ is even. Also,

$$\hat{S}=\frac{1}{N}\sum_{j=1}^{N}\left|X_{j}-\hat{L}\right|.$$

Replace $\hat{L}$ with $L$ if it is known. If $L$ is known then this estimator is distributed as $\left(2N\right)^{-1}S\cdot\chi_{2N}^{2}$ .

$$\begin{eqnarray*} h\left[X\right] & = & \log\left(2e\right)\\ & \approx & 1.6931471805599453094.\end{eqnarray*}$$

Implementation: scipy.stats.laplace