Quantile Regression

The purpose of regression analysis is to expose the relationship between a response variable and predictor variables. In real applications, the response variable cannot be predicted exactly from the predictor variables. Instead, the response for a fixed value of each predictor variable is a random variable. For this reason, we often summarize the behavior of the response for fixed values of the predictors using measures of central tendency. Typical measures of central tendency are the average value (mean), the middle value (median), or the most likely value (mode). Traditional regression analysis is focused on the mean; that is, we summarize the relationship between the response variable and predictor variables by describing the mean of the response for each fixed value of the predictors, using a function we refer to as the conditional mean of the response. The idea of modeling and fitting the conditional-mean function is at the core of a broad family of regression-modeling approaches, including the familiar simple linear-regression model, multiple regression, models with heteroscedastic errors using weighted least squares, and nonlinearregression models. Conditional-mean models have certain attractive properties. Under ideal conditions, they are capable of providing a complete and parsimonious description of the relationship between the covariates and the response distribution. In addition, using conditional-mean models leads to estimators (least squares and maximum likelihood) that possess attractive statistical properties, are easy to calculate, and are straightforward to interpret. Such