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Fisher information and statistical inference for phase-type distributions

Part of: Markov processes Parametric inference

Published online by Cambridge University Press:  14 July 2016

Mogens Bladt ,
Luz Judith R. Esparza  and
Bo Friis Nielsen
Mogens Bladt
Affiliation:
Universidad Nacional Autónoma de México, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, A.P. 20-726, 01000 México DF, Mexico. Email address: bladt@sigma.iimas.unam.mx
Luz Judith R. Esparza
Affiliation:
Technical University of Denmark, Department of Informatics and Mathematical Modeling, Technical University of Denmark, Richard Petersens Plads, Building 305, DK-2800 Kgs. Lyngby, Denmark
Bo Friis Nielsen
Affiliation:
Technical University of Denmark, Department of Informatics and Mathematical Modeling, Technical University of Denmark, Richard Petersens Plads, Building 305, DK-2800 Kgs. Lyngby, Denmark
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Abstract

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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.

Keywords

Phase-type distributionFisher informationEM algorithmNewton–Raphson

MSC classification

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
Type
Part 6. Statistics
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 277 - 293
Copyright
Copyright © Applied Probability Trust 2011 

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