Quantile regression models a conditional quantile of a response, rather than its conditional mean. Fitting by the check loss uses an asymmetric absolute-error penalty, so different values of describe different parts of the response distribution. Nonlinear models may use any identifiable continuous regression function, not only a linear predictor.
If the conditional error distribution function is continuous and strictly increasing with -quantile zero, the conditional expected check loss is uniquely minimized at zero fitted displacement. If every incorrect parameter differs from the true regression function on an event of positive probability, averaging these nonnegative conditional risk differences gives strict identifiability of the true parameter. Integrable errors and bounded regression functions supply finite risks.
The check loss weights positive errors by and negative errors by . It is convex, continuous and Lipschitz with constant . At it is half the absolute error. Its expected value elicits a quantile: asymmetry determines which probability level the fitted location targets.
For an integrable response with continuous strictly increasing distribution function , the population risk for the check loss has derivative . Bounded difference quotients justify differentiation by dominated convergence. Integrating the derivative between the unique quantile and another location proves strict optimality. This proves identifiability without requiring a positive density.
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Quantile regression is a type of regression analysis used in statistics that estimates the relationship between independent variables and specific quantiles (percentiles) of the dependent variable's distribution, rather than just focusing on the mean (as in ordinary least squares regression). This method allows for a more comprehensive analysis of the impact of independent variables across different points in the distribution of the dependent variable.