scholarly journals Inhomogeneous phase-type distributions and heavy tails

2019 ◽  
Vol 56 (4) ◽  
pp. 1044-1064 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Mogens Bladt
Keyword(s):  
Real World ◽  
Heavy Tails ◽  
Heavy Tail ◽  
Transition Rates ◽  
Modelling Framework ◽  
Inhomogeneous Phase ◽  
Fire Insurance ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Heavy Tailed

AbstractWe 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 the carrying over of 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 in modelling a real-world heavy-tailed fire insurance dataset.

scholarly journals Multivariate fractional phase–type distributions

2020 ◽  
Vol 23 (5) ◽  
pp. 1431-1451 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Martin Bladt ◽  
Mogens Bladt
Keyword(s):  
Tail Dependence ◽  
Jump Process ◽  
Multivariate Distributions ◽  
New Class ◽  
Natural Time ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Markovian Jump Process ◽  
Heavy Tailed

Abstract We extend the Kulkarni class of multivariate phase–type distributions in a natural time–fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non–Markovian jump process with ML sojourn times. This new class complements an earlier multivariate ML construction [2] and in contrast to the former also allows for tail dependence. We derive properties and characterizations of this class, and work out some special cases that lead to explicit density representations.

scholarly journals The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey’shTransformation as a Special Case

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Georg M. Goerg
Keyword(s):  
Heavy Tails ◽  
Random Variable ◽  
R Package ◽  
Heavy Tail ◽  
Cumulative Distribution ◽  
Lambert W Function ◽  
Inverse Transformation ◽  
Heavy Tailed ◽  
Probability Density Function Pdf ◽  
Special Case

I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of thisheavy tail Lambert  W × FXrandom variable depends on a tail parameterδ≥0: forδ=0,Y≡X, forδ>0 Yhas heavier tails thanX. ForXbeing Gaussian it reduces to Tukey’shdistribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey’shpdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R packageLambertWimplements most of the introduced methodology and is publicly available onCRAN.

On corrected phase-type approximations of the time value of ruin with heavy tails

2019 ◽  
Vol 36 (1-4) ◽  
pp. 57-75
Author(s):  
Daniel J. Geiger ◽  
Akim Adekpedjou
Keyword(s):  
Heavy Tails ◽  
Risk Model ◽  
Correction Term ◽  
Classical Case ◽  
Penalty Functions ◽  
Tail Behavior ◽  
Time Value ◽  
Phase Type ◽  
Heavy Tailed ◽  
Relative Errors

Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.

scholarly journals PHASE-TYPE DISTRIBUTIONS FOR CLAIM SEVERITY REGRESSION MODELING

2022 ◽  
pp. 1-32
Author(s):  
Martin Bladt
Keyword(s):  
Expectation Maximization ◽  
Regression Models ◽  
Heavy Tails ◽  
Proportional Hazards ◽  
Expectation Maximization Algorithm ◽  
Third Party ◽  
Covariate Information ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Entire Distribution

Abstract This paper addresses the task of modeling severity losses using segmentation when the data distribution does not fall into the usual regression frameworks. This situation is not uncommon in lines of business such as third-party liability insurance, where heavy-tails and multimodality often hamper a direct statistical analysis. We propose to use regression models based on phase-type distributions, regressing on their underlying inhomogeneous Markov intensity and using an extension of the expectation–maximization algorithm. These models are interpretable and tractable in terms of multistate processes and generalize the proportional hazards specification when the dimension of the state space is larger than 1. We show that the combination of matrix parameters, inhomogeneity transforms, and covariate information provides flexible regression models that effectively capture the entire distribution of loss severities.

scholarly journals Gaussian processes with skewed Laplace spectral mixture kernels for long-term forecasting

2021 ◽  
Author(s):  
Kai Chen ◽  
Twan van Laarhoven ◽  
Elena Marchiori
Keyword(s):  
Gaussian Processes ◽  
Multivariate Time Series ◽  
Heavy Tail ◽  
Lottery Ticket ◽  
Spectral Densities ◽  
Heavy Tailed ◽  
Long Term Forecasting ◽  
The Time Domain ◽  
Spectral Mixture

AbstractLong-term forecasting involves predicting a horizon that is far ahead of the last observation. It is a problem of high practical relevance, for instance for companies in order to decide upon expensive long-term investments. Despite the recent progress and success of Gaussian processes (GPs) based on spectral mixture kernels, long-term forecasting remains a challenging problem for these kernels because they decay exponentially at large horizons. This is mainly due to their use of a mixture of Gaussians to model spectral densities. Characteristics of the signal important for long-term forecasting can be unravelled by investigating the distribution of the Fourier coefficients of (the training part of) the signal, which is non-smooth, heavy-tailed, sparse, and skewed. The heavy tail and skewness characteristics of such distributions in the spectral domain allow to capture long-range covariance of the signal in the time domain. Motivated by these observations, we propose to model spectral densities using a skewed Laplace spectral mixture (SLSM) due to the skewness of its peaks, sparsity, non-smoothness, and heavy tail characteristics. By applying the inverse Fourier Transform to this spectral density we obtain a new GP kernel for long-term forecasting. In addition, we adapt the lottery ticket method, originally developed to prune weights of a neural network, to GPs in order to automatically select the number of kernel components. Results of extensive experiments, including a multivariate time series, show the beneficial effect of the proposed SLSM kernel for long-term extrapolation and robustness to the choice of the number of mixture components.

Phase Type Distributions with Finite Support

2014 ◽  
Vol 30 (4) ◽  
pp. 576-597 ◽  
Author(s):  
V. Ramaswami ◽  
N. C. Viswanath
Keyword(s):  
Finite Support ◽  
Phase Type ◽  
Phase Type Distributions

Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

Efficient fitting of long-tailed data sets into phase-type distributions

2002 ◽  
Vol 30 (3) ◽  
pp. 6-8 ◽  
Author(s):  
Alma Riska ◽  
Vesselin Diev ◽  
Evgenia Smirni
Keyword(s):  
Data Sets ◽  
Phase Type ◽  
Phase Type Distributions
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