# Estimate range of parameter

Given a number of points (i.e. [1.1,3.0,0.3,9.9,2.4]), which are random samples of a parameter.

I want to give an estimate about the range, in which the parameter varies. The min/max-range is [0.3,9.9]. However the range could be greater, since i only have samples and not the whole population.

I want to calculate something like "i'm 99%/95% sure that the interval is [0.1,11.1]/[0.2,10.1]".

What do you need for assumptions (i.e. the samples follow normal distribution) for this and how do you calculate it?

I read about Confidence intervals, but from my understanding this is something related, but not the same as i am looking for.

• In general this might be difficult without some sort of parametric assumptions. We know the sample min and max are going to over and under state the true min and max, but it's difficult to say by how much. To see this just imagine cases like this: the true max is very large, but the probability of even being close to that value is very small. You have no way of knowing whether or not you're in one of these scenarios based on your sample. – dsaxton Oct 4 '16 at 15:26
• Great. Can you elaborate more on what "parametric assumptions" are needed, maybe as an answer. You might know some keywords/literature for naming this problem. Thank you very much! – user3613886 Oct 4 '16 at 15:42
• You may find the German Tank Problem relevant. The Wikipedia article <en.wikipedia.org/wiki/German_tank_problem > is a bit technical but all I've got time for now. You might be able to Google something a bit easier. – user20637 Oct 4 '16 at 16:05
• It sounds like a bayesian parameter inference resulting in a credible interval is what you want. Usually one adds some assumptions about the possible measurement outcomes in terms of a model with a prior and then goes on to get a posterior distribution for the parameter of interest from which a credible interval can be derived. Does your example data come from a specific problem/experiment? – Thies Heidecke Oct 4 '16 at 17:29