# Does using bootstrapped samples improve parameter estimates for a fitted distribution?

The R package retimes has a function for fitting an ex-Gaussian distribution to a set of observations. The method involves taking multiple bootstrapped samples of the observations, and fitting the ex-Gaussian to each sample using maximum likelihood estimation. This generates a distribution of parameter values for each of the three parameters of the ex-Gaussian. These distributions are then smoothed, and the mode of each is taken as the best estimate of that parameter of the ex-Gaussian distribution.

My questions are:

1. In what circumstances is this method better than a single maximum likelihood estimate on the observed data?
2. Is there a reference for this technique that would explain the motivation and justify its use? There are several references listed in the documentation for the retimes package, but none of them appear to deal with the bootstrapping technique.
• I haven't read the references for retimes, but I suspect it does not do what you claim. Bootstrapping (especially from small samples) will usually not improve estimates; in fact, if you just use the mean bootstrap estimate, it often will be more biased than the original! (Think about it: if your estimator tends to be, say, 10% below the true value, then the bootstrap estimator will be about 10% lower than the true sample value, making it almost 20% lower than the population value on average.) But if it is used to identify and correct a bias then it may have a point.
– whuber
Jan 20, 2015 at 15:37
• I got in touch with the author of retimes. He said that he has empirical simulation data showing that the method does a good job with small sample sizes. It sounded like the data will be published in the future, so this question will probably go unanswered until then. Jan 29, 2015 at 15:53
• Thank you. Please note that I have not criticized retimes itself: that would be inappropriate, because I have not studied it. I have only noted there is likely a discrepancy between what it does and your characterization of it. One way your question could be answered in the meantime would be for you to review what the software does to check whether your understanding is accurate or needs to be modified.
– whuber
Jan 29, 2015 at 16:44