A family of distributions from the exponential dispersion family with a power-law mean-variance relationship. For power $p$ between 1 and 2, it is a compound Poisson-Gamma distribution that has point mass at zero and is continuous on positive numbers.

Tweedie distributions are distributions from the exponential dispersion family with a power-law mean-variance relationship. For power $p$ between 1 and 2, it is a compound Poisson-Gamma distribution that has point mass at zero and is continuous on positive numbers:

$$\sum_{i=1}^NX_i$$

where $N\sim\text{Pois}(\lambda)$, $X_i\sim\text{Gamma}(\alpha_i,\beta_i)$ and the $X_i$s and $N$ are all independent.

For more on this distribution refer to the wikipedia page or Bent Jorgenson's paper. There is also a book by Jorgenson on the same topic.