# 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?

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.