Uniform Asymptotics for Nonparametric Quantile Regression with an Application to Testing Monotonicity

In this paper, we establish a uniform error rate of a Bahadur representation for local polynomial estimators of quantile regression functions. The error rate is uniform over a range of quantiles, a range of evaluation points in the regressors, and over a wide class of probabilities for observed random variables. Most of the existing results on Bahadur representations for local polynomial quantile regression estimators apply to the fixed data generating process. In the context of testing monotonicity where the null hypothesis is of a complex composite hypothesis, it is particularly relevant to establish Bahadur expansions that hold uniformly over a large class of data generating processes. In addition, we establish the same error rate for bootstrap local polynomial estimators which can be useful for various bootstrap inference. As an illustration, we apply to testing monotonicity of quantile regression and present Monte Carlo experiments based on this example.