Confidence Band for Probability Mass or Density Function

Question

I'm a beginner -- don't hesitate to overexplain.

I need to make inferences about a population distribution from a sample distribution of a continuous variable (which is still useful as an ordinal discrete one).

What I want is a confidence band on the probability function and the right method for getting the density on a given interval (or mass) with a confidence band.

I thought about treating the variable as ordinal-discrete and doing proportion estimates across all the levels (taking care with the 0 "success" ones). Like I said, I'm a beginner, but I sense this is very very wrong.

What's area/method should I explore? It seems like estimated density confidence bands (assuming this is what I need) are fairly recent -- I hope for something mainstream.

Answer 1

You don't need density estimation. I think the answer to your problem is to use nonparametric tolerance intervals. This is very well explained in the book Statistical Intervals by Hahn and Meeker. See the following amazon link: http://www.amazon.com/Statistical-Intervals-Practitioners-Probability-Statistics/dp/0471887692/ref=sr_1_1?s=books&ie=UTF8&qid=1339464464&sr=1-1 . To give a brief description tolerance intervals are intervals that cover at least a specified proportion of the probability distribution with a specified level of confidence. Nonparametric tolerance intervals do this using properties of order statistics.