Suppose I am producing units and have tested the failure time of some proportion of them (say 10%), possibly with some right censoring if they didn't fail within the testing period.
I'm able to fit a Weibull distribution that models the failure time of my units fairly accurately, using MLE to get estimates for the shape and scale parameters. This lets me say, for example, that 8% of my units will fail during the second month of operation.
But I'd like to put a prediction interval around that prediction, to be able to say that 90% of the time, between 3% and 12% of the units will fail during that time. Is there a formula or method for computing these?
(What I've found before asking: I've found formulas for computing confidence intervals. These are much too tight. Far more than 10% of the data falls outside the confidence interval in practice. I've also found papers that discuss prediction intervals around time-to-failure, but what I'm looking for is a pointwise confidence interval for reliability at each possible time interval.)