A bootstrap method for constructing pointwise and uniform confidence bands for conditional quantile functions

Abstract

This paper is concerned with inference about the conditional quantile function in a nonparametric quantile regression model. Any method for constructing a confidence interval or band for this function must deal with the asymptotic bias of nonparametric estimators of the function. In estimation methods such as local polynomial estimation, this is usually done through undersmoothing or explicit bias correction. The latter usually requires oversmoothing. However, there are no satisfactory empirical methods for selecting bandwidths that under- or oversmooth. This paper extends the bootstrap method of Hall and Horowitz (2013) for conditional mean functions to conditional quantile functions. The paper also shows how the bootstrap method can be used to obtain uniform confidence bands. The bootstrap method uses only bandwidths that are selected by standard methods such as cross validation and plug-in. It does not use under- or oversmoothing. The results of Monte Carlo experiments illustrate the numerical performance of the bootstrap method.