ENH: generic support for discontinuous hessian, second derivative of objective, as in quantile regression

[josef-pkt]

usecase quantile regression and robust, RLM

#9182 for quantile regression
see also #1372 (comment) update 2, reference to Hampel et al book. In those cases, discontinuities are handled assuming a (gaussian) reference distribution and not based on the empirical distribution.

M-quantiles have continuous first but discontinuous second derivative. The standard cov_params estimate does not work for M-quantile approximations to quantile regression.
In robust literature this is also the case at least absolute deviation (LAD), i.e. 0.5 quantile.

In those cases the cov_params depends on the density at the discontinuity. This is currently handled with kernel methods in quantile regression, but we cannot immediately reuse it for other models.

(Aside: kinks in psi at other points than zero in RLM norms are at low density region and seem to be negligible.)

The target would be to make the cov_params computation in QuantileRegression available for general or generic M-estimator.

Maybe:
Hessian is needed if we want to use Newton type optimization methods.
For those cases we could use an expected hessian based on the KDE.
(I have not looked at the literature to see how this would work.)