I'm hoping this isn't a ridiculous question! Here goes...
I would like to visualise the range of uncertainty in a probability density function fit to observed data using Maximum Likelihood estimation. I have included a sketch to illustrate what I am aiming for (note this is not based on real data, just a sketch).
The steps I have used so far are:
Use bootstrapping to obtain 1000 samples of my data
Use maximum likelihood estimation to fit a three paramter inverse gamma probability density function (aka Pearson V distribution) to each sample from step 1 .
Taking the median of each parameter value from step 2 to plot the "median" distribution. Ditto for the min, max etc.
However, I think I may be going wrong at step 3, as the median of one parameter value does not necessarily correspond to the median of the other two. Is there a more statistically correct way to go about this?
Thanks in advance for any help - Cross Validated has become my go-to for so many questions over the last year (I just wish I could answer more rather than just ask!).