Resampling methods

Regression quantile estimators solve a linear program and can be computed efficiently. The finite-sample distributions of regression quantiles can be characterized (Koenker, 2005), but they are difficult to use for statistical inference. Suppose that we have data (Formula Presented), where the conditional quantile of Y given X is of interest and assumed to be linear. The asymptotic distributions of regression quantiles are normal under mild conditions, but the asymptotic variance depends on the conditional densities of Y given X = X i $ X=X_i $, which are generally unknown. Statistical inference based on the asymptotic variance is arguably difficult for a simple reason. That is, one needs to use a nonparametric estimate of the asymptotic variance that requires the choice of a smoothing parameter, and such estimates can be quite unstable. Even if the asymptotic variance is well estimated, the accuracy of its approximation to the finite-sample variance depends on the design matrix as well as the quantile level. Resampling methods provide a reliable approach to inference for quantile regression analysis under a wide variety of settings.