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