Heavy-tailed distribution with closed-form ML fit from data Which (if any) heavy-tailed distributions can we compute the maximum likelihood parameters of, given some data to fit the distribution to?
 A: To give a characterization of all such would probably be a very broad question indeed.
However, it's possible to give a strategy to generate a very broad range of examples. 
So here's a strategy to obtain closed form estimators for a large variety of heavy-tailed distributions.
Note the invariance to a broad class of transformations of data (see the second half of that section, not the first part on transformation of the parameter).
So this should allow you to generate as many as you like -
(1) pick some distribution with a nice closed-form MLE for some parameter
(2) pick some nice transformation of the variable (one that retains the invariance to transformations of data) which transforms a random variable with a distribution in (1) to one in the class of heavy-tailed distributions (by whichever definition of heavy-tailedness you like). Exponentiation (or repeated exponentiation, $\exp(\exp(...(x)))$) and inversion are examples of transformations that may sometimes take you to nice heavy-tailed distributions.
(3) voila!
So consider two obvious examples of heavy-tailed distributions with nice closed form for some parameter or parameters - 


*

*The shape parameter (tail index) of a Pareto distribution with lower limit $x_m=1$ (which will clearly be closed form because it corresponds to the rate parameter of an exponential - and indeed it is closed form $\widehat \alpha = \frac{n}{\sum _i \ln x_i}$).

*Both parameters for a lognormal distribution, since the MLEs for the corresponding normal are closed form.
