You can use Michaud's Resampled Efficient Frontier as a technique, or Atillio Meucci's Entropy Pooling.
In Michaud's approach you can sample returns with replacement for each of your assets. Based on these draws you can calculate the expectations, variances, and covariances for each simulation. You can then construct, say, a 1,000 efficient frontiers and use an averaging process to combine them. In my view there are some theoretical flaws with this technique although it tends to work in practice.
Meucci has several case-studies where he implements Entropy Pooling. Under this approach, I would create, say, a 1,000 expected return estimates for your various assets. Each of these can be represented with a view. The probability of each view being correct is 1/1000. You can then blend those views and construct a portfolio using a utility function of your choice (mean-variance, VaR, c-VaR).
Doug Martin has also published exciting work on tail-risk budgeting. If you want to take a portfolio construction approach to handling non-normality his slides are worth taking a look at.
Some interesting related literature for you would be Meucci's Robust Bayesian Allocation which will help you develop a robust optimization solution which I think is your ultimate objective.
How to implement these ideas are discussed elsewhere on the site:
- Entropy Pooling