I want to generate a full density probabilistic forecasting model, where I don't know a priori whether the time series I want to model are intermittent or dense. In both cases, the time series is a count series (i.e. I would rather not have non integer outputs - but I will settle for them if there is no other options).
I know that for intermittent time series, Poisson or Negative binomial are the recommended distributions, and it seems that for dense time series, everybody is just assuming a normal distribution and moving along.
In my case I want a distribution that covers both and is cheap to calculate and parametric - using some non-parametric approach is too computationally expensive (and too painful to code).
My first thought was to simply go with Poisson or NegBin, since for large means, they tended to look like Normal distributions anyway. But then I realized that both of those come with a fixed variance for any given mean, and I want a distribution that can be narrow or wide based on the particular time series in question.
Is there any distribution that satisfies all of my requirements?