# Bootstrap confidence intervals on parameters or on distribution?

Excuse what may be an obvious question about bootstrapping. I got sucked in the Bayesian world early and never really explored bootstrapping as much as I should have.

I ran across an analysis in which the authors were interested in a survival analysis related to some time to failure data. They had about 100 points and used regression to fit a Weibull distribution to the data. A result of this they obtained estimates of the scale and shape parameters. A very traditional approach. However, they next used bootstrapping to sample from the original data set and, for each new sample, performed a regression and came up with a new Weibull distribution. The results of the bootstrapping was then used to construct confidence intervals on the survival distribution.

My intuition is a bit conflicted. I'm familiar with bootstrapping confidence intervals on parameters, but not seen it used for constructing distribution confidence intervals.

Can anyone point me toward a reference/source that might provide some insight? Thanks in advance.

• The question actually is more general than you suppose, because it really doesn't matter how the parameters were estimated. The crux of the matter is that the parameters completely determine the distributions. Thus, a set of simultaneous CIs on the parameters is a CI for the distributions.
– whuber
Apr 15, 2011 at 15:02
• I understand that and perhaps it is that straightforward. Maybe what's nagging at me is that bootstrapping comes with it's own baggage and I was wondering if there was something about the procedure that introduces additional issues when used for this next step. On the other hand, it could just be what I had for breakfast that's gnawing. Thanks for the quick comment. Apr 15, 2011 at 15:51
• Gotta be the breakfast :-)
– whuber
Apr 15, 2011 at 17:23
• There are at least two ways to bootstrap. The simplest way is to simply draw a random sample from the given observations and estimate the model b times (bootstrapping "pairs"). You can also bootstrap using the residuals from a model ("residuals" bootstrap). The first neglects the error structure in the data, which the second method implicitly assumes your model is correct. Efron & Tibshirani (1993). "An Introduction to the Bootstrap" is the place to start. Apr 15, 2011 at 19:58

## 1 Answer

Basically, if you have a joint confidence interval for the parameters that uniquely describe a distribution, then you have a distribution confidence interval. So your problem vanishes... as per whuber's comment.