In reply to Rick Wicklin.

This is the MLE method I had in mind when I wrote my comment https://imgur.com/xU2K09h. It's a bit worse in terms of estimation accuracy compared to the weighted least squares solution for quantile matching estimation but I figured I should share because the more you know. Here's the link to the full code I wrote to estimate how well each method works https://github.com/cats256/assessment-statistics/blob/main/assessment-statistics/flask-server/log_likelihood.py