# Mean of bootstrap maximum likelihood samples?

I am trying to estimate best-fit parameters from some data for a model. For that I use the maximum likelihood estimation (MLE), where eventually I end up finding an estimate of the parameters. In order to get the confidence intervals I use profile likelihood and bootstrapping. However, I was under the impression that the best-fit parameters found by the MLE from the original data is the parameters to use/report. But now I have seen a few articles where they actually provide the mean of parameters found in the bootstrapped samples. This is of course together WITH the true estimates. But does it even make sense to report the mean of the bootstrapped samples, or am I missing anything here ?

I disagree with the initial answer with respect to the claim that the mean of the bootstraps is in general a redundant quantity because it only is under certain conditions and these depend strongly on the model and parameters you are considering.

To give my answer some context, let's set up an example: If we estimate a linear model via ML over a multivariate normal distribution, the MLEs have certain favorable features: They are asymptotically unbiased, consistent, efficient and - relevant for CI computation - normally distributed – if your data are really drawn from a multivariate normal distribution.

If multivariate normal distribution holds and your sample is large, MLEs and ML-CIs are the best thing you could ever report (with respect to the listed properties). Reporting bootstrap CIs can have mainly two reasons, either the distributional assumption is violated or it is just too complicated to derive the second derivative of the likelihood to get analytical standard errors.

Assuming that the distributional assumption is violated, the properties of the MLE are not guaranteed any more. Sometimes it is possible to derive the MLEs without the distributional assumption (for instance, the MLEs happen to be equivalent to the least square estimates in linear regression). Thus, you might know your MLEs are unbiased and consistent irrespective of the distribution. However, the distribution of MLEs across samples may then be unknown. Bootstrapping is an easy way out of this problem.

There are many approaches to get CIs from a bootstrap and some of them perform poorly if the distribution across bootstrap samples is not nearly symmetric or normal. The mean of the bootstrap samples is a straightforward indicator of not so well-behaved bootstrap distributions because, e.g., skewed bootstrap distributions generally result in a difference between the original estimate and the mean across bootstraps. You could also turn the argument around: If the two are close, it is likely that the simplest ways of computing bootstrap CIs suffice (and that bootstrapping is not generally inappropriate). I have once had a reviewer who wanted me to report the means across bootstraps for this very reason.

To be clear, I am not proposing that the mean of the bootstrap samples is the ideal indicator of potential problems or that it must always be reported. All I am trying to point out is that it is not - in general - a redundant quantity.

I attach some further readings: Bootstrap based bias correction

https://garstats.wordpress.com/2018/01/23/bias-correction/

When is the bootstrap estimate of bias valid?

The mean of the bootstrap sample is a biased estimator of non-pivotal quantities. You can correct the bias by generating a 95% CI around the bootstrapped mean, then shifting the upper and lower limits by the difference between the bootstrap mean and the MLE. It does not make sense to report the mean of a bootstrap sample, except to illustrate the bias if it's a methods oriented paper

So probably they report it because, via bootstrapping, they want to reporduce the empirical distribution of the estimator in repeated samples. And they plot the mean of such empirical distribution, for the sake of completion. So that it is a bit easier to visualize the reason why the confidence interval is the one reported. In other words, they want to give a sense of how close the estimate is to the mean of the empirical distribution.

But in my opinion, it is something absolutely redundant that can be omitted. In the vast majority of the cases I have seen up to now, indeed, this figure is not reported. This is my personal experience, clearly. But I think I have almost never seen it. Because clearly it can be omitted, and in my opinion for clarity it is far better to omit, otherwise you put too much stuff in front of the reader. In my opinion, if anybody wants to get the confidence interval, the best thing is just to tell the reader what the interval is and omit info that are not strictly necessary: the simpler, the clearer, the better.