Is it possible to correctly use regularized linear or generalized linear models with the mob function in partykit?
I imagine the null hypothesis of the parameter stability tests that partykit uses would need to change to a non-standard Brownian Bridge because the score function is no longer 0 on the full set of data points, but rather shifted by the gradient of the penalizing term.
Yes, that is correct. However, I've never worked through this in detail and what really would need to be modified to obtain a clean functional central limit theorem.
One ad hoc modification would be to simply substract the mean score from the individual score contributions so that these sum to zero again. My gut feeling is that this should make the resulting test slightly more conservative so that at least you are on the "right side".
Another solution would be to use the conditional inference procedures from ctree() rather than the parametric stability tests from mob(). You can simply supply the estfun(model(...)) as the ytrafo. Feel free to get in touch with us (Torsten, Heidi, and me) by direct e-mail for more guidance and discussion.