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

Abdelhakim Ziani, András Horváth, Paolo Ballarini

TL;DR

This paper tackles the challenge of modeling heavy-tailed distributions by proposing a hybrid mixture of Bernstein Phase-Type (BPH) and Hyperexponential (HE) distributions, blending BPH’s body accuracy with HE’s tail flexibility. A gradient-based optimization (Adam) automates and stabilizes the HE tail initialization, addressing prior sensitivity and manual tuning issues. The resulting BPH_HE model demonstrates superior performance over standalone BPH or HE models across multiple distributions and is validated in queuing scenarios where tail behavior materially impacts performance metrics. The approach offers a practical, analytically tractable framework for accurate heavy-tailed approximation with direct applicability to risk-sensitive and systems analysis tasks.

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.

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

TL;DR

This paper tackles the challenge of modeling heavy-tailed distributions by proposing a hybrid mixture of Bernstein Phase-Type (BPH) and Hyperexponential (HE) distributions, blending BPH’s body accuracy with HE’s tail flexibility. A gradient-based optimization (Adam) automates and stabilizes the HE tail initialization, addressing prior sensitivity and manual tuning issues. The resulting BPH_HE model demonstrates superior performance over standalone BPH or HE models across multiple distributions and is validated in queuing scenarios where tail behavior materially impacts performance metrics. The approach offers a practical, analytically tractable framework for accurate heavy-tailed approximation with direct applicability to risk-sensitive and systems analysis tasks.

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.

Paper Structure

This paper contains 13 sections, 26 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Heavy tailed distributions
  5. Phase type distributions
  6. Bernstein phase type distributions
  7. Hyperexponential approximation of heavy-tailed distributions
  8. Methodology
  9. Approximation with mixture of BPH and HE distributions
  10. Optimizing data initialization for the HE tail approximation
  11. Experimental setup
  12. Queuing system experiment
  13. Conclusion and future work

Figures (6)

  • Figure 1: A degree 3 PH distribution
  • Figure 2: Bernstein PH approximation of a CDF $F(x)$.
  • Figure 3: PDF resulting from a CDF-based BPH approximation of the Burr distribution with parameters $c=2, d=1$.
  • Figure 4: PDF of the HE approximation of a Burr distribution
  • Figure 5: PDF of BPH_HE approximation of a Burr distribution (a, b) and a Pareto distribution (c, d). The Burr case corresponds to Figures \ref{['fig:bph_burr']} and \ref{['fig:he_burr']}.
  • ...and 1 more figures