Laplace (Double Exponential, Bilateral Exponential) Distribution¶
f(x)=12e−|x|F(x)={12exx≤01−12e−xx>0G(q)={log(2q)q≤12−log(2−2q)q>12
md=mn=μ=0μ2=2γ1=0γ2=3
The ML estimator of the location parameter is
ˆL=median(Xi)
where Xi is a sequence of N mutually independent Laplace RV’s and the median is some number between the 12Nth and the (N/2+1)th order statistic ( e.g. take the average of these two) when N is even. Also,
ˆS=1NN∑j=1|Xj−ˆL|.
Replace ˆL with L if it is known. If L is known then this estimator is distributed as (2N)−1S⋅χ22N .
h[X]=log(2e)≈1.6931471805599453094.
Implementation: scipy.stats.laplace