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.