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Heavy-tailed phase-type distributions: a unified approach.

Martin Bladt1, Jorge Yslas2

  • 1Faculty of Business and Economics, University of Lausanne, Quartier de Chambronne, Lausanne, 1015 Switzerland.

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This study unifies heavy-tailed phase-type distributions, offering flexible models for various tail behaviors. New multivariate extensions and explicit algorithms are provided for practical applications.

Keywords:
Frailty modelsHeavy tailsParameter estimationPhase-typeScale mixtures

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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Statistical Modeling

Background:

  • Phase-type distributions model time until absorption in Markov jump processes.
  • Regular phase-type distributions offer mathematical tractability and a hidden Markov structure interpretation.
  • Existing extensions allow for heavy tails but lack a unified framework.

Purpose of the Study:

  • To present a unifying theory for heavy-tailed phase-type distributions.
  • To provide flexible models capturing diverse tail behaviors.
  • To introduce novel multivariate extensions and practical algorithms.

Main Methods:

  • Development of a unifying theoretical framework for phase-type distributions.
  • Incorporation of discrete/continuous scaling, fractional-time semi-Markov processes, and time-inhomogeneous Markov processes.
  • Construction of multivariate extensions analogous to matrix frailty models.

Main Results:

  • A unified model encompassing existing heavy-tailed phase-type distribution approaches.
  • Demonstration of the model's ability to capture various tail behaviors beyond heavy tails.
  • Development of explicit Expectation-Maximization (EM) algorithms for parameter estimation.
  • Illustration of model performance with synthetic and real-life data.

Conclusions:

  • The unified theory provides a versatile and comprehensive approach to phase-type distributions.
  • The proposed models and algorithms facilitate practical application in statistical modeling.
  • The multivariate extensions offer new avenues for analyzing complex dependent data structures.