Expansions for the distribution and quantiles of a regular functional of the empirical distribution with applications to nonparametric confidence intervals (Q788418)

In this paper power series in \(n^{-1}\) are established for the cumulants of \(T(F_ n)\), where \(T(\cdot)\) is a real-valued sufficiently smooth functional on \({\mathbb{R}}^ s\) and \(F_ n\) denotes the empirical distribution function of a random sample \(X_ 1,...,X_ n\) of size n from a distribution F on \({\mathbb{R}}^ s\). The coefficients in the power series for the cumulants are - up to a remainder \({\mathcal O}(n^{-2})\)- expressed in terms of the (Gateaux) derivatives of T(F). The results are applied to the problem of obtaining an expansion for the distribution of a suitable normalized \(T(F_ n).\) Also accurate confidence intervals for T(F) are established with the aid of these expansions. The results are illustrated by examples: the author discussed the case of the sample variance and the one-sample Student t- statistic in detail. The validity of the expansions in this paper are checked with the aid of the results of \textit{R. N. Bhattacharya} and \textit{J. K. Ghosh}, Ann. Stat. 6, 434-451 (1978; Zbl 0396.62010). The emphasis in the present paper is on the explicit computation of the coefficients in the expansions for the cumulants of \(T(F_ n)\).