I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical inversion that is quite convincing. Then, i tried through fitdistrplus to estimate the resulting density by MLE on weibull, gamma, pareto and burr models, which produced the following plots:
It look like the density that i am looking for lives on the positive real axis, has only one mode, and behave quite nicely.
My goal is now to recover an analytic expression for the coresponding CDF (at least). I tried doing the Broomwitch integration as this wikipedia page suggests, but i am not skilled enough in complex integration to make it through...
Do you think that an analytic expression could be recovered for this CDF ? How should i proceed ?
Thanks for your input.