{"ID":2870052,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.14428","arxiv_id":"2509.14428","title":"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.","short_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 un...","url_abs":"https://arxiv.org/abs/2509.14428","url_pdf":"https://arxiv.org/pdf/2509.14428v1","authors":"[\"Haolin Zou\",\"Heyuan Yao\",\"Victor de la Peña\"]","published":"2025-09-17T21:00:17Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.CO\"]","methods":"[]","has_code":false}