I'm trying to figure out the most robust way to estimate the error on the best-fit parameters of an exponential model ($y = x^a$) for some data. I have heard that bootstrapping is a solid way to attempt this and so have fit models to 1000+ random samples of the data (generated with replacement) and saved the best-fit $a$ for each sample.
The problem is that the distribution of these bootstrapped values looks extremely skewed and is nowhere close to Gaussian. I know that with sample statistics like mean, standard error, etc the Central Limit Theorem would imply that the distribution of results approximates a Gaussian as the sample size increases. I don't think this would apply to model parameters, so I suppose it is not super surprising that the bootstrapped distribution is not.
I'm not sure how to express the uncertainty on my best-fit estimate of $a$ since typical measures like standard error wouldn't apply. Is there some sort of established method to estimate this sort of error on the peak of a skewed distribution? Thank you for any help.