Why don't people use nonparametric methods to get around the problem of assuming certain parameters are "nicely" distributed? (Not a physicist, but curious – this seems like the "obvious" solution.)
In physics experiments they want to fix the structure of the model and know the assumptions. They want to know the distribution and parameters to hold. If assumptions don't hold, they must find out why, find better assumptions and fix the model.
To say it differently: physicists are not trying to discover statistical laws. They are trying to discover physical laws trough statistics.
Is there a concrete reason we can't be naive and just bootstrap confidence intervals for example? Of course I defer to the physicists here – but I'm curious whether there's some simple high-level reason the usual tricks don't work.
See also: https://twitter.com/pietrovischia/status/1512174848558219270
This does not seems like bleeding edge, it seems like "Gaussian approximations for everything."
In fact, this is the reference for the technique they used: https://cds.cern.ch/record/183996/files/OUNP-88-05.pdf. (Although the criticism seems to be leveled at the way they estimate the correlations, not that linear estimator specifically?)