Quantile Regression

AbstractClassical least squares regression may be viewed as a natural way of extending the idea of estimating an unconditional mean parameter to the problem of estimating conditional meanfunctions; the crucial link is the formulation of an optimization problem that encompasses both problems. Likewise, quantile regression offers an extension of univariate quantile estimation to estimation of conditional quantile functions via an optimization of a piecewise linear objective function in the residuals. Median regression minimizes the sum of absolute residuals, an idea introduced by Boscovich in the eighteenth century, and developed by Edgeworth in the nineteenth century.The asymptotic theory of quantile regression closely parallels the theory of the univariate sample quantiles. Computation of quantile regression estimators may be formulated as a linear programming problem and efficiently solved by simplex or barrier methods. A close link to rank‐based inference has been forged from the theory of the dual regression quantile process, or regression rankscore process. Recent work has extended quantile regression into time‐series, spatial models, survival analysis, and nonparametric estimation.