Quantile regression is a statistical technique designed to estimate, and conduct inference about, conditional quartile functions. Classical linear regression methods based on minimizing sums of squared residuals enable researchers to estimate models for conditional mean functions, quartile regression methods offer a mechanism for estimating models for the conditional median function, and the full range of other conditional quartile functions. Quantile regression has been used in a broad range of application settings. Reference growth curves for quartiles of childrens' height and weight have a long history in pediatric medicine. Quantile regression methods may be used to estimate upper and lower quartile reference curves as a function of age, sex, and other covariates without imposing stringent parametric assumptions on the relationships among these curves. Quantile regression methods have been widely used in economics to study determinants of wages, discrimination effects, and trends in income inequality. In ecology, theory often suggests how observable covariates affect limiting sustainable population sizes, and quartile regression has been used to directly estimate models for upper quartiles of the conditional distribution rather than inferring such relationships from models based on conditional central tendency. In survival analysis, and event history analysis more generally, there is often also a desire to focus attention on particular segments of the conditional distribution, for example survival prospects of the oldest-old, without the imposition of global distributional assumptions, suggesting that quartile regression may be appropriate. This project will focus primarily on problems of statistical inference in quartile regression. The approach is based on an ingenious suggestion by Khmaladze for dealing with problems of testing composite null hypotheses involving unknown nuisance parameters based on the empirical distribution function. Khmaladze's martingale transformation approach provides a general strategy for purging the effect of the estimated nuisance parameters from the first order asymptotic representation of the empirical process and thereby restores the feasibility of "asymptotically distribution free" tests. The approach seems especially attractive in quantile regression settings and should be capable of greatly expanding the scope of inference currently described in the econometrics literature.
