REGRESSION QUANTILES' A simple minimization problem yielding the ordinary sample quantiles in the location model is shown to generalize naturally to the linear model generating a new class of statistics we term "regression quantiles." The estimator which minimizes the sum of absolute residuals is an important special case. Some equivariance properties and the joint asymptotic distribution of regression quantiles are established. These results permit a natural generalization to the linear model of certain well-known robust estimators of location. Estimators are suggested, which have comparable efficiency to least squares for Gaussian linear models while substantially out-performing the least-squares estimator over a wide class of non-Gaussian error distributions.

We say that a student scores at the th quantile of a standardized exam if he performs better than the proportion of the reference group of students and worse than the proportion (1-). Thus, half of students perform better than the median student and half perform worse. Similarly, the quartiles divide the population into four segments with equal proportions of the reference population in each segment. The quintiles divide the population into five parts; the deciles into ten parts. The quantiles, or percentiles, or occasionally fractiles, refer to the general case. Quantile regression as introduced by Koenker and Bassett (1978) seeks to extend these ideas to the estimation of conditional quantile functions-models in which quantiles of the conditional distribution of the response variable are expressed as functions of observed covariates.In Figure 1, we illustrate one approach to this task based on Tukey's boxplot (as in McGill, Tukey and Larsen, 1978). Annual compensation for the chief executive officer (CEO) is plotted as a function of firm's market value of equity. A sample of 1,660 firms was split into ten groups of equal size according to their market capitalization. For each group of 166 firms, we compute the three quartiles of CEO compensation: salary, bonus and other compensation, including stock options (as valued by the Black-Scholes formula at the time of the grant). For each group, the bow-tie-like box represents the middle half of the salary distribution lying between the first and third quartiles. The horizontal line near the middle of each box represents the median compensation for each group of CEOs, and the y

No abstract

The penalized least squares interpretation of the classical random effects estimator suggests a possible way forward for quantile regression models with a large number of "fixed effects". The introduction of a large number of individual fixed effects can significantly inflate the variability of estimates of other covariate effects. Regularization, or shrinkage of these individual effects toward a common value can help to modify this inflation effect. A general approach to estimating quantile regression models for longitudinal data is proposed employing 1 regularization methods. Sparse linear algebra and interior point methods for solving large linear programs are essential computational tools.

Abstract. Quantile regression is an evolving body of statistical methods for estimating and drawing inferences about conditional quantile functions. An implementation of these methods in the R language is available in the package quantreg. This vignette offers a brief tutorial introduction to the package. R and the package quantreg are open-source software projects and can be freely downloaded from CRAN: http://cran.r-project.org.