Asymptotic Representations of Skewness Estimators of Studentized $ U $-Statistics

Abstract

A skewness is a measure of symmetry of a distribution and appears in an Edgeworth expansion of a standardized or studentized statistic. It has been found in simulation studies that jackknife estimators of the skewness have downward biases. Fujioka and Maesono (2000) have obtained a normalizing transformation with residual term o(n −1) and they pointed out that in order to construct the normalizing transformation, we need an asymptotic representation of a skewness estimator. Maesono (1998) has obtained the asymptotic representation of the jackknife skewness estimators and discussed their biases. In this paper we propose another skewness estimator of a U-statistic and obtain asymptotic representations of both estimators with remainder term o p (n −1) and discuss the biases theoretically.

Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

“…Asymptotic expansions for symmetric statistics are studied by many people. See, e.g., Callaert-Janssen-Veraverbeke (1980) [1], Withers (1988) [2], Maesono (2004) [3], and so on. They treat U-statistics with non-degenerate kernels.…”