# Probability distribution models compatible with quantile regression

I am familiarizing myself with quantile regression. I understand it is first and foremost an estimation method such as e.g. OLS. But I wonder about the probability distribution models for which quantile regression makes sense.

To use an analogy, the OLS estimator $$\hat\beta^{OLS}:=(X^\top X)^{-1}X^\top y$$ is a minimum variance linear unbiased estimator when the true probability distribution model is \begin{align} y &= X\beta+\varepsilon, \\ \varepsilon &\sim d(0,\sigma^2) \end{align} or in other words, $$y\mid X\sim d(X\beta,\sigma^2)$$, where $$d$$ is some (unspecified) probability distribution.
Moreover, $$\hat\beta^{OLS}$$ is a maximum likelihood estimator when the probability model is \begin{align} y &= X\beta+\varepsilon, \\ \varepsilon &\sim N(0,\sigma^2) \end{align} or in other words, $$y\mid X\sim N(X\beta,\sigma^2)$$. So in some sense, the OLS estimator "naturally implies" the above probability distribution models.

Question 1: What could we say about the quantile estimator $$\hat\beta^{QR}_{\tau}:= \arg\min_{\beta}\sum_{i=1}^{n}\rho_{\tau}(y_i-X_i\beta)$$ where $$\rho_{\tau}$$ is the quantile loss function (and the superscript $$^{QR}$$ stands for "quantile regression")? Does it "naturally imply" a probability distribution model for $$y\mid X$$?

Question 2: If we assumed particular true conditional quantiles $$\beta^{QR}_{\tau}:= \arg\min_{\beta}\mathbb{E}\left(\rho_{\tau}(y-X\beta)\right)$$ (where $$\beta_{\tau}$$ may differ across different values of $$\tau$$) for a continuum of quantiles between 0 and 1, we would get an implicit conditional probability model for $$y\mid X$$.
But could the model be expresed explicitly in a nice way? If so, I would welcome a simple example.

P.S. In principle, the question covers more than just linear quantile regressions, but for practical purposes an answer addressing just the linear case would suffice.

• In books I've read, it's been treated as a non parametric regression but apparently the quantile estimator is the maximum likelihood estimator of an asymmetric double exponential distribution. See here web1.sph.emory.edu/users/hwu30/teaching/statcomp/Notes/… Dec 28, 2018 at 19:50
• @machazthegamer, interesting. For a single quantile level $\tau$, your comment (slightly expanded) could serve as an answer. For multiple quantiles $\tau_i$ (a finite number of them) with different corresponding true $\beta^{QR}_{\tau_i}$, the model is probably underdetermined. Dec 28, 2018 at 21:55
• @machazthegamer, I would like to accept your answer if you post your extended comment as one. Jan 15, 2019 at 16:49
• Could someone expound on what was said here? Jun 23, 2020 at 3:18
• @RylanSchaeffer, where exactly? Jun 23, 2020 at 5:44