Advanced Estimation of Higher-Order Cumulants in Bi-Additive Statistical Models for Applied Sciences

This work introduces improved estimators for fourth-order cumulants in bi-additive models involving both fixed and random effects. Cumulants are powerful statistical measures that extend beyond traditional moments, offering more nuanced descriptions of probability distributions, including characteristics such as skewness and kurtosis. By utilizing the cumulant-generating function and least-squares methods, we develop advanced estimators that enable accurate inference from independent and identically distributed (i.i.d.) data. The proposed bi-additive models, which incorporate deterministic fixed components and independent random terms, facilitate precise estimation of fourth-order cumulants while accounting for the variability commonly observed in real-world data. These models also enable the systematic analysis of distributions with parameters related to location, dispersion, and shape, thus shedding light on their underlying structure. We demonstrate that the proposed methods offer both theoretical robustness and practical utility across diverse applied contexts. We illustrate the applications of this methodology in two domains: anomaly detection in sensor networks, where higher-order cumulants help identify deviations from expected patterns, and variability analysis in materials science, where they capture subtle differences in material properties. Our results emphasize the significance of higher-order cumulants in revealing complex features in data that are often obscured in mean-variance-based analyses. This contribution emphasizes both the theoretical significance and practical benefits of adopting advanced cumulant estimators in bi-additive models, equipping researchers in the applied sciences with novel tools for exploring and understanding the complexity of empirical distributions.