Edgeworth Expansion and Normalizing Transformation of Ratio Statistics and Their Application

Abstract

Some statistics in common use take the form of a ratio of two statistics, such as sample correlation coefficient, Pearson's coefficient of variation, cumulant estimators, and so on. In this article, using an asymptotic representation of the ratio statistics, we will obtain an Edgeworth expansion and a normalizing transformation with remainder term o(n −1/2). The Edgeworth expansion is based on a Studentized ratio statistic, which is studentized by a consistent variance estimator. Applying these results to the sample correlation coefficient, we obtain the normalizing transformation and an asymptotic confidence interval of the correlation coefficient without assuming specific underlying distribution. This normalizing transformation is an extension of the Fisher's z-transformation.