Saddlepoint tests for quantile regression

Abstract

AbstractQuantile regression is a flexible and powerful technique which allows us to model the quantiles of the conditional distribution of a response variable given a set of covariates. Regression quantile estimators can be viewed as M‐estimators and standard asymptotic inference is readily available based on likelihood‐ratio, Wald, and score‐type test statistics. However these statistics require the estimation of the sparsity function where G and g are the cumulative distribution function and the density of the regression errors, respectively, and this can lead to nonparametric density estimation. Moreover the asymptotic distribution for these statistics can provide an inaccurate approximation of tail probabilities and this can lead to inaccurate P‐values, especially for moderate sample sizes. Alternative methods which do not require the estimation of the sparsity function include rank techniques and resampling methods to obtain confidence intervals, which can be inverted to test hypotheses. These are typically more accurate than the standard M‐tests. In this article we show how accurate tests can be obtained by using a nonparametric saddlepoint test statistic. The proposed statistic is asymptotically distributed, does not require the specification of the error distribution, and does not require the estimation of the sparsity function. The validity of the method is demonstrated through a simulation study, which shows both the robustness and the accuracy of the new test compared to the best available alternatives. The Canadian Journal of Statistics 44: 271–299; 2016 © 2016 Statistical Society of Canada