Some parts of our research fall out of our expertise. We thought about creating a Kaggle challenge just for this part. But we lack fund resources for the reward, so we could not offer any money. We are thinking now about offering co-authorship in a paper, do you think this would work? Otherwise, which other things would you offer as reward in Kaggle apart from money?
It's true that fitting the Kaggle competition framework is a bit of a constraint, but if you understand that framework and see how your problem fits into it, I'd suggest you e-mail email@example.com with a description of the data & problem (and ideally sample data), even if you don't have funding. We can see what comes of it!
(Disclaimer: I work for Kaggle.)
This sounds to me like the textbook definition of a potential collaboration opportunity. I would speak with researchers in the computer science, machine learning, mathematics, and/or statistics departments and try to establish a long-term collaborative relationship. I would also speak with other faculty in your department to see how they handle this problem, which they likely have as well. It may be as simple as paying a grad or reasonably bright undergrad a few bucks an hour to crunch numbers. (Actually, you shouldn't take for granted that the grad student is reasonably bright... but I digress.)
This will be beneficial in both the short- and long-term, in a number of ways. In the short term, you'll hopefully get to analyze your data and generate a publication. In the long term, it's very likely that you'll need such expertise again, and you'll have the resources available for the analysis. Additionally, you'll find that having someone with analytics expertise on hand when devising whatever study you plan on doing can be immeasurably useful, as they will help you determine what measurements need to be taken and what data needs to be collected to ensure that you can get the most out of your dataset.
Some competitions have "Kudos" (or "Jobs", or "Knowledge") as a prize, see Kaggle competitions sorted from the smallest prizes. I don't how it does work with proposing a competition (maybe it is not free).
I guess if the problem itself fits in the Kaggle framework (which is a pretty strong constraint), and you can try it for free (I don't know if it is the case), then why not. Especially as even two "Kudos" competitions attract 50 and 153 teams, respectively. Just it is on you to make it as attractive as possible.
Of course, intellectual contribution requires a co-authorship. But if I had the right expertise, I would run for it :).