An asymptotic representation of a ratio of two statistics and its applications

Some statistics in common use take a form of a ratio of two statistics.In this paper, we will discuss asymptotic properties of the ratio statistic.We obtain an asymptotic representation of the ratio with remainder term o _p(n ^-1) and a Edgeworth expansion with remainder term o(n ^-1/2) And as example, the asymptotic representation and the Edgeworth expansion of the jackknife skewness estimator for U-statistics are established and we discuss the biases of the skewness estimator theoretically.We also apply the result to an estimator of Pearson’s coefficient of variation and the sample correlation coefficient.