Quantile Regression › Summary

Quantile regression replaces squared error with the pinball loss $\rho_\tau(u) = u(\tau - \mathbf{1}\{u < 0\})$, whose population minimizer is the $\tau$-quantile rather than the mean. Restricting the conditional quantile to be linear in the predictors, $Q_{Y \mid X=x}(\tau) = x^\top \beta(\tau)$, turns estimation into minimizing an average of piecewise linear terms. That objective is convex but non-differentiable, so we reformulate it as a linear program with $2n + p + 1$ unknowns, splitting each residual into non-negative positive and negative parts. The dual of that program exposes its bounded response score at fixed predictors: every observation's multiplier is capped between $\tau - 1$ and $\tau$, and the first-order condition $X^\top a = 0$ reduces, at the intercept, to the counting rule that the negative-residual count must bracket $n\tau$ after including zero residuals.

The payoff is a description of several parts of the conditional distribution rather than one mean. Fitting several levels produces a fan whose width is a local measure of spread, which lets us estimate variation in conditional spread directly. A conditional-mean model can also be combined with a separate variance model. The costs include the median-regression efficiency penalty under the regular Gaussian conditions described earlier, potentially low precision when conditional density near a cutoff is sparse, depending also on $\tau$ and the predictor design, no general OLS-style fixed linear map from responses to coefficients, and quantile curves that can cross because each level is estimated independently. Crossing is diagnosed by counting order violations on a grid and repaired by rearrangement, which sorts fitted values across levels at each covariate point and removes crossings by construction.

The most consequential caution is that a nominal band is not a guarantee. Reporting "90 percent" is a claim about held-out coverage, and the only honest way to support it is to measure coverage on data that were not used to fit the model, exactly as we did with the per-level calibration plot. A misspecified form, sparse information near an extreme quantile, or a new target population can invalidate a nominal label. Quantile regression alone supplies no general finite-sample coverage guarantee; methods such as split conformal prediction require separate calibration data and assumptions such as exchangeability. Part XII currently lists planned chapters on calibration and prediction uncertainty. Within those limits, quantile regression complements the conditional-mean models in Part V by targeting selected quantiles without specifying a full response distribution, in the same language we will need when the response becomes a time to an event in the chapters that follow.