The Informational Content in Lepto-Variance and Its Relation to Higher Moments

Abstract:Lepto-regression is defined as the machine learning process of constructing a Regression Tree of a target feature on itself. It is a novel, model-free method potentially revealing information on important sample structure properties. But it is yet not clear what the informational content of lepto-variance is and how it is related to other well-known statistics of a sample. One significant finding is that 58% of the historical US stock return variability is 1-bit lepto-variance that can not be explained by any financial factor. The central question investigated in this paper is to use small normal N(0, 1) drawn samples to explore how the 1-bit sample lepto-variance and lepto-ratio relate to sample variance, skewness and excess kurtosis. Using a large sample simulation, the lepto ratio of a normal is found to converge to 36.3%. For smaller normally distributed simulated N(0, 1) samples, while lepto-variance itself is highly correlated to sample variance, lepto-variance as a fraction of total variance is highly correlated to excess kurtosis. Both lepto-variance and lepto-ratio are orthogonal to sample skew. Another finding is that while lepto-ratio is strongly correlated to lepto-variance it remains orthogonal to sample variance.

Methodology (stat.ME); Machine Learning (cs.LG); Risk Management (q-fin.RM); Machine Learning (stat.ML)

arXiv:2609.25144 [stat.ME]

(or arXiv:2609.25144v1 [stat.ME] for this version)

https://doi.org/10.48550/arXiv.2609.25144 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

https://doi.org/10.4236/ib.2025.173013 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.