I would like to estimate the parameters associated with a distribution following the Laplace distribution.
I have created experimental data following the Laplace distribution, using the following code:
data = RandomVariate[ExponentialPowerDistribution[1, 0.5, 1], 1000];
Given the data, I found the shape, location, and scale parameters by using FindDistributionParameters
:
FindDistributionParameters[data, ExponentialPowerDistribution[Kappa, Mu, Sigma]]
The obtained result was {1.11375, 0.504605, 1.10716}.
But, if my understanding is correct, the Laplacian distribution is not differentiable at "Mu" (the single peak of the distribution), which gives me trouble understanding the estimation results because MLE assumes the differentiability of the distribution function. So, my questions are:
(Q1) Are the above estimation results reliable (reasonable, or plausible)?
(Q2) If the answer to (Q1) is yes, what is the algorithm used in FindDistributionParameters dealing with the non-differentiability of the distribution function?
(Q3) If the answer to (Q1) is no, then, is there any reasonable way to come up with the parameter estimation for the above example? (In fact, I would like to obtain the standard error of each parameter in a reasonable way too.)
Thank you very much for your help in advance!