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

Abstract

Let $T(\cdot)$ be a suitably regular functional on the space of distribution functions, $F$, on $R^s$. 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 $T(F_n)$ where $F_n$ 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 $1 - \alpha + O(n^{-j/2})$ for any given $j$. Examples include approximate nonparametric confidence intervals for the mean and variance of a distribution on $R$.