Fitting Phase-type Distributions via the EM Algorithm

Scandinavian Journal of Statistics

Volumn 23, Issue 4, 1996, Pages 419-441

  Asmussen, Søren ^b   Nerman, Olle ^a   Olsson, Marita ^b  

^a UNIVERSITY OF GOTHENBURG   (Sweden)

^b NONE

Author keywords

Coxian distribution; Density estimation; EM algorithm; Hidden Markov chain; I divergence; Phase type distributions; Relative entropy

Indexed keywords

EID: 0030502888     PISSN: 03036898     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (50)

1. Aalen, O. O. (1943). Phase-type distributions: computer algebra and a simple mixing model. Manuscript, University of Oslo.
2. Aalen, O. O. (1995). On phase type distributions in survival analysis. Scand. J. Statist. 22, 447-463.
3. Alberg, A. (1961). Estimating the infinitesimal generator of a continuous time, finite state Markov process. Ann. Math. Statist. 38, 727-753.
4. Asmussen, S. (1992). Phase-type representations in random walls and queueing problems. Ann. Probab. 20, 772-789.
5. Asmussen, S. & Bladt, M. (1996). Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix-analytic methods in stochastic models (eds A. S. Alfa & S. Chakravarty), Marcel Dekker, New York (to appear).
6. Asmussen, S. & Nerman, O. (1991). Fitting phase-type distributions via the EM algorithm. Symposium i Anvendt Statistik, Copenhagen, January 21-23, 1991 (ed. K. Vest Nielsen), 335-346. UNI-C, Copenhagen.
7. Asmussen, S. & Rolski, T. (1991). Computational methods in risk theory: a matrix-algorithmic approach. Insurance: Math. Econom. 10, 259-274.
8. Basawa, I. V. & Rao, B. L. S. (1980). Statistical inference for stochastic processes. Academic Press, London.
9. Baum, L. E., Pertrif, T., Soules, G. & Weiss, N. (1970). A maximisation technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41, 164-171.
10. Bobbio, A., Cumani, A., Prernoli A. & Saracco (1980). Modelling and identification of non-exponential distributions by homogeneous Marikov processes. Proc. 6th Adv. Reliab. Symp., Bradford, pp. 373-392.
11. Bobbio, A. & Cumani, A. (1990). ML estimation to the parameters of a PH distributions in triangular canonical form. In Computer performance evaluation (eds G. Balbo & G. Serazzi), 33-46, Elsevier, Amsterdam.
12. Bobbio, A. & Telek, M. (1994). A benchmark for PH estimation algorithms: results for acyclic PH. Commun. Statist. Stochastic Models 10, 661-667.
13. Bux, W. & Herzog, U. (1977). The phase concept: approximation of measured data and performance analysis. In Computer performance (eds K. M. Chandy & M. Reiser), 23-38. North-Holland, Amsterdam.
14. Cox, D. R., (1953). A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Philos. Soc. 51, 313-319.
15. Cumani, A. (1982). On the canonical representation of homogeneous Markov processes modelling failure-time distributions. Microelectron. Reliab. 22, 583-602.
16. Dehon, M. & Latouch, G. (1982). A geometric interpretation of the relations between the exponential and the generalized Erlang distributions. Adv. Appl. Probab. 14, 885-897.
17. Dempster, A. P., Laird, N. M. & Rubin D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B. 39, 1-38.
18. Faddy, M. J. (1990). Compartmental models with phase-type residence time distributions. Appl. Stochastic Models Data Anal. 6, 121-127.
19. Faddy, M. J. (1993). A structured compartmental model for drug kinetics. Biometrics 49, 243-248.
20. Faddy, M. J. (1994). Examples of fitting structured phase type distributions. Appl. Stochastic Models Data Anal. 10, 247-256.
21. Harris, C. M, & Sykes, E. A. (1987). Likelihood estimation for generalized mixed exponential distributions. Naval Res. Logist. Quart. 34, 251-279.
22. Hjort, N. L. (1992). On inference in parametric survival data models. Int. Statist. Rev. 60, 355-387.
23. Hoem, J. M. (1969). Purged and partial Markov chains. Skand. Aktuarietidskr. 56, 147-155.
24. Häggström, O., Asmussen, S. & Nerman, O. (1992). EMPHT - A program for fitting phase-type distributions. Technical report, Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden.
25. Jamshidian, M. & Jennrich, R. I. (1993). Conjugate gradient acceleration of the EM algorithm. J. Amer. Statist. Assoc. 88, 221-228.
26. Johnson, M. A., (1990). Selecting parameters of phase distributions: combining nonlinear programming, heuristics and Erlang distributions. Technical Report.
27. Johnson, M. A. & Taaffe, M. R. (1989). Matching moments to phase distributions: mixtures of Erlang distributions of common order. Commun. Statist. Stochastic Models 5, 711-743.
28. Johnson, M. A. & Taaffe, M. R., (1990a). Matching moments to phase distributions: nonlinear programming approaches. Commun. Statist. Stochastic Models 6, 259-281.
29. Johnson, M. A. & Taaffe, M. R. (1990b). Matching moments to phase distributions: density function shapes. Commun. Statist. Stochastic Models 6, 283-306.
30. Jonsson, E., Andersson, M. & Asmussen, S. (1994). A practical dependability measure for degradable computer systems with non-exponential degradation. To appear in the proceedings of Safeprocess '94, Helsinki, June 1994.
31. Kao, E. P. C. (1988). Computing the phase-type renewal and related functions. Technometrics 30, 87-93.
32. Kullback, S. (1978). Information theory and statistics. Peter Smith, Gloucester, MA.
33. Lang, A. & Arthur J. L. (1994). Parameter approximation for phase-type distributions. Technical Report, Oregon State University, Corvallis.
34. Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov chains. Stochastic Process. Applic. 40, 127-143.
35. Lipsky, L. (1992). Queueing theory - a linear algebraic approach. Macmillan, New York.
36. Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44, 226-233.
37. Meilijson, I. (1989). A fast improvement to the EM algorithm on its own terms. J. Roy. Statist. Soc. Ser. B 51, 127-138.
38. Moler, C. & Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801-836.
39. Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models. Johns Hopkins University Press, Baltimore, MD.
40. Ó Cinnéide, C. A. (1987). On non-uniqueness of representations of phase-type distributions. Commun. Statist. Stochastic Models 5, 247-259.
41. Olsson, M. (1996). Estimation of phase type distributions from censored data. Scand. J. Statist. 23, 443-460.
42. Orchard, T. & Woodbury, M. A. (1972). A missing information principle: theory and applications. Proc. 6th Berkeley Symposium on Math. Statist. and Probab. 1, 697-715.
43. Redner, R. A. & Walker, H. F. (1984). Mixture densities, maximum likelihood and EM algorithm. SIAM Rev. 26, 195-237.
44. Ruhe, A. (1980). Fitting empirical data by positive sums of exponentials. Siam. J. Sci. Statist. Comput. 1, 481-498.
45. Ryden, T. (1993). Parameter estimation for Markov modulated Poisson processes and overload control of spc switches. PhD thesis, Department of Mathematical Statistics, Lund Institute of Technology, Sweden.
46. Sengupta, B. (1989). Markov processes whose steady-state distribution is matrix-exponential with an application to the GI/G/1 queue. Adv. Appl. Probab. 21, 159-180.
47. Silverman, B. W. (1986). Density estimation for statistics and data analysis. Chapman & Hall, New York.
48. Sundberg, R. (1974). Maximum likelihood theory for incomplete data from an exponential family. Scand. J. Statist. 1, 49-58.
49. Sundberg, R. (1976). An iterative method for solution of the likelihood equations for incomplete data from exponential families. Commun. Statist. Simulation Computation B5, 55-64.
50. Wu, C. F. J. (1983). On the convergence properties of the EM algorithm. Ann. Statist. 11, 95-103.