Journal of Applied Probability

Fisher information and statistical inference for phase-type distributions

Mogens Bladt, Luz Judith R. Esparza, and Bo Friis Nielsen


This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton--Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton--Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.

Article information

J. Appl. Probab., Volume 48A (2011), 277-293.

First available in Project Euclid: 18 October 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes

Phase-type distribution Fisher information EM algorithm Newton--Raphson


Bladt, Mogens; Esparza, Luz Judith R.; Nielsen, Bo Friis. Fisher information and statistical inference for phase-type distributions. J. Appl. Probab. 48A (2011), 277--293. doi:10.1239/jap/1318940471.

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