Calculating nonparametric confidence intervals for quantiles using fractional order statistics

In this paper, we provide an easy-to-program algorithm for constructing the preselected 100(1 - α)% nonparametric confidence interval for an arbitrary quantile, such as the median or quartile, by approximating the distribution of the linear interpolation estimator of the quantile function Q(L)(u) = (1 - ε)X([n'u]) + εX([n'u]+1) with the distribution of the fractional order statistic Q(I)(u) = X(n'u), as defined by Stigler, where n' = n + 1 and [·] denotes the floor function. A simulation study verifies the accuracy of the coverage probabilities. An application to the extreme-value problem in flood data analysis in hydrology is illustrated.