I was thinking of this problem, and I'm not sure if I'm right with this approach.
X is a R.V. with unknown distribution, bounded to the interval [a,b], with a < b and both finite. If I take a very large sample of this population, separate training and testing dataset (the latter with n samples) train a model (keeping things general, because I don't want to use any property of it) to make predictions to future samples, what is the prediction interval (not the confidence interval) for a single data point ?
I thought that because X is bounded, the maximum variance of the population is $$(b-a)^2/4$$ so the (1-α)100% prediction interval for a data point wouldn't be larger than: $$ \left(\hat{x} - t_{1-\alpha/2, n-1}*\frac{|b-a|}{2}*{\sqrt{{\frac{n+1}{n}}}}, \hat{x} + t_{1-\alpha/2, n-1}*\frac{|b-a|}{2}*{\sqrt{{\frac{n+1}{n}}}}\right) $$
That's basically the equation for the Prediction Interval if X is Normal, but could I use it even if X isn't ?
Should I assume a distribution for the population ?
Maybe I could make tighter bounds with Confidence Intervals of the sample variance, but I wanted to make things simpler. I would also appreciate any references for non-parametric prediction intervals