Generalized quantile regression? Transforming a conditional quantile like we transform conditional expected value Linear models are models like $\mathbb E\left[Y\vert X\right]=X\beta$.
Linear quantile regression models replace $\mathbb E\left[Y\vert X\right]$ with $Q_{\tau}\left(Y\vert X\right)$ for conditional quantile $\tau$, giving $Q_{\tau}\left(Y\vert X\right)=X\beta$.
Generalized linear models include a link function to transform the conditional expected value of a linear model, so $g\left(\mathbb E\left[Y\vert X\right]\right)=X\beta$.
Is there any role for generalized quantile regression that transforms the conditional quantile $\tau$, like $g\left(Q_{\tau}\left(Y\vert X\right)\right)=X\beta$? How would we go about developing a likelihood function to maximize or a loss function to minimize?

Bounded distributions.
We have linear probability model that comes with a drawback of producing fitted probabilities that are outside of the [0,1] interval for some values of the regressor $X$. We have a generalized linear model – logistic regression – that fixes that.
Suppose we want to model the quantiles of a bounded distribution such as $\text{Uniform}(0,1)$. By using a linear model, we would face the same type of problem as described above. It would be nice to have the fitted quantiles inside of the [0,1] interval. This is where generalized quantile regression of the logistic type (but not only) would come in handy.
I have no thoughts about maximum likelihood estimation as of now.
