Abstract
The inclusion of generally distributed random variables in stochastic models is often tackled by choosing a parametric family of distributions and applying fitting algorithms to find appropriate parameters. A recent paper proposed the approximation of probability density functions (PDFs) by Bernstein exponentials, which are obtained from Bernstein polynomials by a change of variable and result in a particular case of acyclic phase-type distributions. In this paper, we show that this approximation can also be applied to cumulative distribution functions (CDFs), which enjoys advantageous properties; by focusing on CDFs, we propose an approach to obtain stochastically ordered approximations.
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References
Aldous, D., Shepp, L.: The least variable phase type distribution is erlang. Stoch. Models 3(3), 467–473 (1987)
Asmussen, S., Nerman, O.: Fitting phase-type distributions via the EM algorithm. In: Proceedings of Symposium i Advent Statistik, Copenhagen, pp. 335–346 (1991)
Baccelli, F., Makowski, A.M.: Multidimensional stochastic ordering and associated random variables. Oper. Res. 37(3), 478–487 (1989)
Bobbio, A., Cumani, A.: ML estimation of the parameters of a PH distribution in triangular canonical form. In: Computer Performance Evaluation, pp. 33–46. Elsevier (1992)
Bobbio, A., Horváth, A., Telek, M.: Matching three moments with minimal acyclic phase type distributions. Stoch. Models 21, 303–326 (2005)
Feldman, A., Whitt, W.: Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Perform. Eval. 31, 245–279 (1998)
Fourneau, J.M., Pekergin, N.: A numerical analysis of dynamic fault trees based on stochastic bounds. In: Campos, J., Haverkort, B.R. (eds.) QEST 2015. LNCS, vol. 9259, pp. 176–191. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22264-6_12
Horváth, A., Telek, M.: Approximating heavy tailed behavior with phase-type distributions. In: Proceedings of 3rd International Conference on Matrix-Analytic Methods in Stochastic models, Leuven, Belgium (2000)
Horváth, A., Vicario, E.: Construction of phase type distributions by Bernstein exponentials. In: Iacono, M., Scarpa, M., Barbierato, E., Serrano, S., Cerotti, D., Longo, F. (eds.) EPEW ASMTA 2023. LNCS, vol. 14231, pp. 201–215. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-43185-2_14
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM (1999)
Neuts, M.: Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin, pp. 173–206. University of Louvain (1975)
Neuts, M.F.: Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore (1981)
Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer, New York (2003)
Riska, A., Diev, V., Smirni, E.: An EM-based technique for approximating long-tailed data sets with PH distributions. Perform. Eval. 55(1–2), 147–164 (2004)
Rivlin, T.J.: An Introduction to the Approximation of Functions. Courier Corporation (1981)
Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9–12), 1153–1168 (2007)
Acknowledgments
M. Telek was supported by the OTKA K-138208 project of the Hungarian Scientific Research Fund. E. Vicario was supported bt the RESTART project, as a part of the Italian PNNR programme.
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Horváth, A., Horváth, I., Paolieri, M., Telek, M., Vicario, E. (2024). Approximation of Cumulative Distribution Functions by Bernstein Phase-Type Distributions. In: Hillston, J., Soudjani, S., Waga, M. (eds) Quantitative Evaluation of Systems and Formal Modeling and Analysis of Timed Systems. QEST+FORMATS 2024. Lecture Notes in Computer Science, vol 14996. Springer, Cham. https://doi.org/10.1007/978-3-031-68416-6_6
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