I have a question regarding interpretation of prediction intervals and confidence intervals. The definitions I've seen is:
(1) A prediction interval for a single future observation is an interval that will, with a specified degree of confidence, contain the next (or some other prespecified) randomly selected observation from a population.
(2) A tolerance interval is an interval that one can claim to contain at least a specified proportion, p, of the population with a specified degree of confidence
My question is whether these definitions are equivalent to
(1) A prediction interval gives point estimates for the quantiles of a distribution. So for example, a 95% confidence interval gives a point estimate for the 0.025 and 0.975 quantiles of a distribution.
(2) A tolerance interval gives (one side of) a confidence interval estimate for a quantile. So for example, a 95% tolerance level for a 90% of the values gives a 1-sided confidence interval at 95% for the 0.05 and 0.95 quantiles?
