“Exact” Bootstrap Confidence Bands for the Quantile Function via Steck's Determinant

Abstract

This note introduces a new small sample method for producing nonparametric confidence bands for a quantile function based upon a single confidence coefficient obtained via Steck's determinant. The procedure is an “exact” bootstrap method (EB) in the sense that it does not require any Monte Carlo simulations, only an empirical “plug-in” estimator of the quantile function. We compare this new method to the classical method based upon empirical quantile processes, as well as with the bootstrap version of another empirical process based approach. The EB method produces confidence bands with correct coverage probabilities for samples as low as size n = 10 over a wider range of quantiles as compared to the classical approach. Furthermore, the standard bootstrap approach to the problem has poor coverage probabilities, thus illustrating the utility of the EB method.