Unbiased Estimation of Central Moments in Unbalanced Two- and Three-Level Models

2026-02-03

AI Summary

This study addresses the systematic bias inherent in conventional estimators of central moments within multilevel random-effects models characterized by unbalanced group sizes, which impedes accurate characterization of higher-order distributional features. Building upon multilevel linear modeling theory and leveraging algebraic properties of expectation operators alongside combinatorial mathematics, the authors derive, for the first time, closed-form unbiased estimators for higher-order central moments under two- and three-level unbalanced designs. Specifically, the proposed estimators cover second- through fourth-order moments for two-level models and second- through third-order moments for three-level models, with formulations tailored to both group-level and observation-level averaging schemes. This work fills a critical theoretical gap in unbiased estimation of higher-order moments in unbalanced multilevel settings and substantially enhances the accuracy of higher-order statistical inference.

Technology Category

Reasoning under Uncertainty: Graphical ModelsMachine Learning: Calibration & Uncertainty QuantificationCognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and rankingWeb Mining and Content Analysis: Models for Web evolution

Abstract

This paper derives closed-form unbiased estimators of central moments in multilevel random-effects models with unbalanced group sizes. In a two-level model, we provide unbiased estimators for the second, third, and fourth central moments under both group-level and observation-level averaging. In a three-level model, we provide unbiased estimators for the second and third central moments.

Problem

• unbiased estimation
• central moments
• unbalanced multilevel models
• random-effects models

Innovation

• unbiased estimation
• central moments
• multilevel models
• unbalanced data
• random-effects models