Expansions for the Distribution and Quantiles of a Regular Functional of the Empirical Distribution with Applications to Nonparametric Confidence Intervals

Let T () be a suitably regular functional on the space of distribution functions, F, on Rˢ. A method is given for obtaining the derivatives of T at F. This is used to obtain asymptotic expansions for the distribution and quantiles of T (Fₙ) where Fₙ is the empirical distribution of a random sample of size n from a distribution F with an absolutely continuous component. One- and two-sided confidence intervals for T (F) are given of level 1 - + O (n^-j/2) for any given j. Examples include approximate nonparametric confidence intervals for the mean and variance of a distribution on R.