# What methods to use to estimate fat tailed distributions (practical considerations)

My problem is to estimate the distribution some observed data. I am exploring what best distribution to use. The data is fat tailed most likely following a power law in the tails (the burr distribution has the maximal likelihood among a set of usual distribution)

I would like to produce an invertible (on $\mathbb{R}^+$) CDF function from observed data. It's important for the CDF to be invertible, that the inverse be fast to calculate (can be achieved computationally) and that I can use on all the domain $\mathbb{R}^+$. Also, the storing parameters or serialization is an important consideration.

Since I need it inversible and full domain usage => Empirical is not possible.

This leaves me with testing several parametric distributions or use a kernel density estimation.

For parametric estimation:

• Which method should I use, given the fat tails? in R's fitdistrplus package they have maximum likelihood (mle), moment matching (mme), quantile matching (qme) or maximizing goodness-of-fit estimation (mge).

For nonparametric density kernel estimation - again given the fat tails:

1. Does it estimate the power law tail correctly?
2. Will it work on the whole domain even far outside what i have in the observations, or would i have to extend it ?
3. Is it easy to store?