Inhomogeneous phase-type distributions and heavy tails

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset.

[1]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[2]  M. Denuit,et al.  Composite Lognormal–Pareto model with random threshold , 2011 .

[3]  Christian Max Møller.  Numerical evaluation of Markov transition probabilities based on the discretized product integral , 1992 .

[4]  M. Bladt,et al.  Calculation of ruin probabilities for a dense class of heavy tailed distributions , 2015 .

[5]  Gordon E. Willmot,et al.  On the Class of Erlang Mixtures with Risk Theoretic Applications , 2007 .

[6]  Ren Asmussen,et al.  Fitting Phase-type Distributions via the EM Algorithm , 1996 .

[7]  G. S. Goodman,et al.  Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Lothar Breuer.  From Markov jump processes to spatial queues , 2003 .

[9]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[10]  Richard D. Gill,et al.  Product Integration , 2001 .

[11]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[12]  Tom Reynkens,et al.  Modeling Censored Losses Using Splicing: A Global Fit Strategy with Mixed Erlang and Extreme Value Distributions , 2016, 1608.01566.

[13]  Mogens Bladt,et al.  Matrix-Exponential Distributions in Applied Probability , 2017 .

[14]  J. Teugels,et al.  Reinsurance: Actuarial and Statistical Aspects , 2017 .

[15]  V. Ramaswami,et al.  A New Class of Models for Heavy Tailed Distributions in Finance and Insurance Risk , 2012 .

[16]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[17]  R. Gill,et al.  A Survey of Product-Integration with a View Toward Application in Survival Analysis , 1990 .

[18]  J. C. Helton,et al.  Numerical approximation of product integrals , 1976 .

[19]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[20]  David P. M. Scollnik.  On composite lognormal-Pareto models , 2007 .

[21]  M. Bladt,et al.  Fitting phase–type scale mixtures to heavy–tailed data and distributions , 2017, Extremes.