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arXiv · 2509.14428

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

Haolin Zou·Heyuan Yao·Victor de la Peña
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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.

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Haolin Zou, Heyuan Yao, Victor de la Peña. 2025-09-17. A Scalable Formula for the Moments of a Family of Self-Normalized Statistics. https://arxiv.org/abs/2509.14428

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