Approximating Heavy-Tailed Distributions with a Mixture of Bernstein Phase-Type and Hyperexponential Models

2025-10-30

AI Summary

Existing heavy-tailed distribution modeling suffers from inaccurate fitting in the bulk region, insufficient characterization of tail decay, and poor robustness and bulk adaptability of hyperexponential (HE) models due to sensitivity to initial parameters. To address these issues, this paper proposes a hybrid modeling framework integrating Bernstein phase-type (BPH) and hyperexponential (HE) distributions. The BPH component ensures high-accuracy approximation in the bulk region, while the HE component provides superior flexibility for modeling slowly decaying tails. A moment-matching-based robust initialization strategy is further designed to enhance the convergence stability of the HE estimation. Extensive queueing-theoretic simulations demonstrate that the proposed hybrid model significantly outperforms standalone BPH or HE models across key metrics—including mean, coefficient of variation, and tail probabilities—achieving unified, high-fidelity modeling of the entire heavy-tailed distribution, encompassing both the non-tail (bulk) and slow-decay tail regions.

Technology Category

• Planning, Routing, and Scheduling: Model-Based Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Heuristic Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Heavy-tailed distributions, prevalent in a lot of real-world applications such as finance, telecommunications, queuing theory, and natural language processing, are challenging to model accurately owing to their slow tail decay. Bernstein phase-type (BPH) distributions, through their analytical tractability and good approximations in the non-tail region, can present a good solution, but they suffer from an inability to reproduce these heavy-tailed behaviors exactly, thus leading to inadequate performance in important tail areas. On the contrary, while highly adaptable to heavy-tailed distributions, hyperexponential (HE) models struggle in the body part of the distribution. Additionally, they are highly sensitive to initial parameter selection, significantly affecting their precision. To solve these issues, we propose a novel hybrid model of BPH and HE distributions, borrowing the most desirable features from each for enhanced approximation quality. Specifically, we leverage an optimization to set initial parameters for the HE component, significantly enhancing its robustness and reducing the possibility that the associated procedure results in an invalid HE model. Experimental validation demonstrates that the novel hybrid approach is more performant than individual application of BPH or HE models. More precisely, it can capture both the body and the tail of heavy-tailed distributions, with a considerable enhancement in matching parameters such as mean and coefficient of variation. Additional validation through experiments utilizing queuing theory proves the practical usefulness, accuracy, and precision of our hybrid approach.

Problem

• Modeling heavy-tailed distributions with slow tail decay
• Combining BPH and HE models to capture body and tail
• Optimizing initial parameters to enhance robustness and accuracy

Innovation

• Hybrid model combining BPH and HE distributions
• Optimization sets initial parameters for HE component
• Captures both body and tail of heavy-tailed distributions