Fitting with matrix-exponential distributions

Stochastic Models

Volumn 21, Issue 2-3, 2005, Pages 377-400

  Fackrell, Mark ^a  

^a UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

Matrix exponential distribution; Maximum likelihood estimation; Phase type distribution; Rational laplace stieltjes transform; Semi infinite programming

Indexed keywords

MAXIMUM LIKELIHOOD ESTIMATION; PARAMETER ESTIMATION;

MATRIX-EXPONENTIAL DISTRIBUTIONS; PHASE-TYPE DISTRIBUTION; SEMI-INFINITE PROGRAMMING;

MATRIX ALGEBRA;

EID: 27944498125     PISSN: 15326349     EISSN: 15324214     Source Type: Journal    
DOI: 10.1081/STM-200056227     Document Type: Conference Paper

References (32)

1. Asmussen, S.; Bladt, M. Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix-analytic Methods in Stochastic Models, Chakravarthy, S.R., Alfa, A.S., Eds.; Lecture Notes in Pure and Applied Mathematics; Marcel Dekker: New York, 1997; Vol. 183, 313-341.
2. Asmussen, S.; Nerman, O.; Olsson, M. Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 1996, 23, 419-441.
3. Asmussen, S.; O'Cinneide, C.A. Matrix-exponential distributions. In Encyclopedia of Statistical Science Update, Kotz, S., Read, C.B., Banks, D.L., Ed.; John Wiley and Sons: New York, 1998; Vol. 2, 435-440.
4. Bobbio, A.; Cumani, A. ML Estimation of the parameters of a PH distribution in triangular canonical form. In Computer Performance Evaluation; Balbo, G., Serazzi, G., Eds.; Elsevier Science Publishers, 1992; 33-46.
5. Botta, R.B.; Harris, C.M.; Marchal, W.G. Characterizations of generalized hyperexponential distribution functions. Communications in Statistics - Stochastic Models 1987, 3, 115-148.
6. Bux, W.; Herzog, U. The phase concept: Approximation of measured data and performance analysis. In Computer Performance; Chandy, K.M., Reiser, M., Eds.; North Holland Publishing Company, 1977; 23-38.
7. Chong, E.K.P.; Zak, S.H. An Introduction to Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization; John Wiley and Sons: New York, 1996.
8. Cohen, J.W. The Single Server Queue, Volume 8 of North-Holland Series in Applied Mathematics and Mechanics; North-Holland Publishing Company: Amsterdam, 1969.
9. Cox, D.R. A use of complex probabilities in the theory of stochastic processes. Proceedings of The Cambridge Philosophical Society 1955, 51, 313-319.
10. Debnath, L. Integral Transforms and their Applications; CRC Press: Boca Raton, 1995.
11. Dehon, M.; Latouche, G. A geometric interpretation of the relations between the exponential and generalized Erlang distributions. Advances in Applied Probability 1982, 14, 885-897.
12. Fackrell, M. Characterization of Matrix-Exponential Distributions. PhD thesis, School of Applied Mathematics, University of Adelaide, South Australia, 2003.
13. Faddy, M.J. Compartmental models with phase-type residence-time distributions. Applied Stochastic Models and Data Analysis 1990, 6, 121-127.
14. Faddy, M.J. A structured compartmental model for drug kinetics. Biometrics 1993, 49, 243-248.
15. Faddy, M.J. Examples of fitting structured phase-type distributions. Applied Stochastic Models and Data Analysis 1994, 10, 247-255.
16. Faddy, M.J. On inferring the number of phases in a Coxian phase-type distribution. Communications in Statistics - Stochastic Models 1998, 14, 407-417.
17. Faddy, M.J.; McClean, S.I. Analysing data on lengths of stay of hospital patients using phase-type distributions. Applied Stochastic Models in Business and Industry 1999, 15, 311-317.
18. Hampel, K. Modelling Phase-type Distributions. Honours thesis, Department of Statistics, University of Adelaide, South Australia, 1997.
19. Harris, C.M.; Marchal, W.G. Distribution estimation using Laplace transforms. INFORMS Journal on Computing 1998, 10, 448-458.
20. Harris, C.M.; Sykes, E.A. Likelihood estimation for generalized mixed exponential distributions. Naval Research Logistics 1987, 34, 251-279.
21. Hettich, R; Kortanek, K.O. Semi-infinite programming: Theory, methods, and applications. SIAM Review 1993, 35, 380-429.
22. Hórvath, A.; Telek, M. Approximating heavy tailed behaviour with phase type distributions. In Advances in Algorithmic Methods for Stochastic Models; Latouche, G., Taylor, P., Eds.; Notable Publications: New Jersey, USA, 2000; 191-213.
23. Johnson, M.A. Selecting parameters of phase distributions: Combining nonlinear programming, heuristics, and Erlang distributions. ORSA Journal on Computing 1993, 5, 69-83.
24. Latouche, G.; Ramaswami, V. Introduction to Matrix Analytic Methods in Stochastic Models; SIAM: Philadelphia, 1999.
25. Lipsky, L.; Ramaswami, V. A Unique Minimal Representation of Coxian Service Centres, Technical Report; Department of Computer Science and Engineering, University of Nebraska, Lincoln, 1985.
26. Moran, P.A.P. An Introduction to Probability Theory, Oxford Scientific Publications, Clarendon Press: Oxford, 1984.
27. Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach; The John Hopkins University Press: Baltimore, 1981.
28. O'Cinneide, C.A. Phase-type distributions and invariant polytopes. Advances in Applied Probability 1991, 23, 515-535.
29. O'Cinneide, C.A. Triangular order of triangular phase-type distributions. Communications in Statistics - Stochastic Models 1993, 9, 507-529.
30. Reemtsen, R; Görner, S. Numerical methods for semi-infinite programming: A survey. In Semi-Infinite Programming, Reemtsen, R., Rückmann, J-J., Eds.; Volume 25 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers: Boston, 1998; 195-275.
31. Schmickler, L. MEDA: Mixed Erlang distributions as phase-type representations of empirical distribution functions. Communications in Statistics - Stochastic Models 1992, 8, 131-156.
32. Smith, W.L. On the distribution of queueing times. Proceedings of the Cambridge Philosophical Society 1953, 49, 449-461.