A Scalable Formula for the Moments of a Family of Self-Normalized Statistics

Abstract: Following the student t-statistic, normalization has been a widely used method in statistic and other disciplines including economics, ecology and machine learning. We focus on statistics taking the form of a ratio over (some power of) the sample mean, the probabilistic features of which remain unknown. We develop a unified formula for the moments of these self-normalized statistics with non-negative observations, yielding closed-form expressions for several important cases. Moreover, the complexity of our formula doesn't scale with the sample size $n$. Our theoretical findings, supported by extensive numerical experiments, reveal novel insights into their bias and variance, and we propose a debiasing method illustrated with applications such as the odds ratio, Gini coefficient and squared coefficient of variation.