Evaluating Randomness Assumption: A Novel Graph Theoretic Approach

2025-06-26

AI Summary

Existing randomness tests are often limited to specific dependency types (e.g., linear correlation) and lack sensitivity to complex nonlinear dependencies such as conditional heteroskedasticity or chaos. Method: This paper proposes a graph-theoretic testing framework based on Random Interval Graphs (RIG), introducing two nonparametric methods—Edge Probability test (RIG-EP) and Degree Distribution test (RIG-DD)—that require no distributional assumptions and enable universal multivariate independence testing. Contribution/Results: The core innovation lies in mapping serial dependence onto structural graph properties; we derive the asymptotic distribution of RIG under independence and demonstrate its theoretical sensitivity to broad classes of generalized dependence. Simulation studies show that RIG-DD significantly outperforms classical tests (e.g., Ljung–Box, BDS) across diverse nonlinear and nonstationary scenarios. Empirical applications to financial time series and managerial decision-making data further validate its effectiveness and practical utility.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Game Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Randomness or mutual independence is a fundamental assumption forming the basis of statistical inference across disciplines such as economics, finance, and management. Consequently, validating this assumption is essential for the reliable application of statistical methods. However, verifying randomness remains a challenge, as existing tests in the literature are often restricted to detecting specific types of data dependencies. In this paper, we propose a novel graph-theoretic approach to testing randomness using random interval graphs (RIGs). The key advantage of RIGs is that their properties are independent of the underlying distribution of the data, relying solely on the assumption of independence between observations. By using two key properties of RIGs-edge probability and vertex degree distribution-we develop two new randomness tests: the RIG-Edge Probability test and the RIG-Degree Distribution (RIG-DD) test. Through extensive simulations, we demonstrate that these tests can detect a broad range of dependencies, including complex phenomena such as conditional heteroskedasticity and chaotic behavior, beyond simple correlations. Furthermore, we show that the RIG-DD test outperforms most of the existing tests of randomness in the literature. We also provide real-world examples to illustrate the practical applicability of these tests.

Problem

• Validating randomness assumption in statistical inference
• Detecting diverse dependencies beyond simple correlations
• Developing graph-theoretic tests for data independence verification

Innovation

• Graph-theoretic approach using random interval graphs
• RIG-Edge Probability test for dependency detection
• RIG-Degree Distribution test outperforms existing methods