Improved Quantile Inference Via Fixed-Smoothing Asymptotics And Edgeworth Expansion

Estimation of a sample quantile's variance requires estimation of the probability density at the quantile. The common quantile spacing method involves smoothing parameter m. When m, n → ∞ , the corresponding Studentized test statistic asymptotically follows a standard normal distribution. Holding m fixed asymptotically yields a nonstandard distribution dependent on m that contains the Edgeworth expansion term capturing the variance of the quantile spacing. Consequently, the fixed-m distribution is more accurate than the standard normal under both asymptotic frameworks. For the fixed-m test, I propose an m to maximize power subject to size control, as calculated via Edgeworth expansion. Compared with similar methods, the new method controls size better and maintains good or better power in simulations. Results for two-sample quantile treatment effect inference are given in parallel.