PHASE-TYPE DISTRIBUTIONS FOR CLAIM SEVERITY REGRESSION MODELING

Regression Models ◽

Heavy Tails ◽

Proportional Hazards ◽

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.

Improvement of expectation-maximization algorithm for phase-type distributions with grouped and truncated data

Applied Stochastic Models in Business and Industry ◽

10.1002/asmb.1919 ◽

2012 ◽

Vol 29 (2) ◽

pp. 141-156 ◽

Cited By ~ 13

Author(s):

Hiroyuki Okamura ◽

Tadashi Dohi ◽

Kishor S. Trivedi

Keyword(s):

Truncated Data ◽

Phase Type ◽

Phase Type Distributions

Multivariate finite-support phase-type distributions

Journal of Applied Probability ◽

10.1017/jpr.2020.65 ◽

2020 ◽

Vol 57 (4) ◽

pp. 1260-1275

Author(s):

Celeste R. Pavithra ◽

T. G. Deepak

Keyword(s):

Distribution Function ◽

Finite Support ◽

Marginal Distributions ◽

New Class ◽

Phase Type ◽

Phase Type Distributions ◽

Support Phase

AbstractWe introduce a multivariate class of distributions with support I, a k-orthotope in $[0,\infty)^{k}$ , which is dense in the set of all k-dimensional distributions with support I. We call this new class ‘multivariate finite-support phase-type distributions’ (MFSPH). Though we generally define MFSPH distributions on any finite k-orthotope in $[0,\infty)^{k}$ , here we mainly deal with MFSPH distributions with support $[0,1)^{k}$ . The distribution function of an MFSPH variate is computed by using that of a variate in the MPH $^{*} $ class, the multivariate class of distributions introduced by Kulkarni (1989). The marginal distributions of MFSPH variates are found as FSPH distributions, the class studied by Ramaswami and Viswanath (2014). Some properties, including the mixture property, of MFSPH distributions are established. Estimates of the parameters of a particular class of bivariate finite-support phase-type distributions are found by using the expectation-maximization algorithm. Simulated samples are used to demonstrate how this class could be used as approximations for bivariate finite-support distributions.

Inhomogeneous phase-type distributions and heavy tails

Journal of Applied Probability ◽

10.1017/jpr.2019.60 ◽

2019 ◽

Vol 56 (4) ◽

pp. 1044-1064 ◽

Cited By ~ 3

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.

Multiple scaled contaminated normal distribution and its application in clustering

Statistical Modelling ◽

10.1177/1471082x19890935 ◽

2019 ◽

pp. 1471082X1989093 ◽

Cited By ~ 1

Author(s):

Antonio Punzo ◽

Cristina Tortora

Keyword(s):

Heavy Tails ◽

Simulated Data ◽

Principal Component ◽

Multivariate Normal ◽

Degree Of Contamination ◽

Robust Estimates ◽

Directional Detection ◽

Heavy Tailed

The multivariate contaminated normal (MCN) distribution represents a simple heavy-tailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers (also referred to as ‘bad’ points herein) and automatically detect bad points. The price of these advantages is two additional parameters: proportion of good observations and degree of contamination. However, in a multivariate setting, only one proportion of good observations and only one degree of contamination may be limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution. Among its parameters, we have an orthogonal matrix Γ. In the space spanned by the vectors (principal components) of Γ, there is a proportion of good observations and a degree of contamination for each component. Moreover, each observation has a posterior probability of being good with respect to each principal component. Thanks to this probability, the method provides directional robust estimates of the parameters of the nested MN and automatic directional detection of bad points. The term ‘directional’ is added to specify that the method works separately for each principal component. Mixtures of MSCN distributions are also proposed, and an expectation-maximization algorithm is used for parameter estimation. Real and simulated data are considered to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.

An Asymmetric Distribution with Heavy Tails and Its Expectation–Maximization (EM) Algorithm Implementation

Symmetry ◽

10.3390/sym11091150 ◽

2019 ◽

Vol 11 (9) ◽

pp. 1150 ◽

Cited By ~ 1

Author(s):

Neveka M. Olmos ◽

Osvaldo Venegas ◽

Yolanda M. Gómez ◽

Yuri A. Iriarte

Keyword(s):

Heavy Tails ◽

Real Data ◽

Independent Random Variables ◽

Asymmetric Distribution ◽

Data Sets ◽

Normal Distributions ◽

The Moment ◽

Algorithm Implementation

In this paper we introduce a new distribution constructed on the basis of the quotient of two independent random variables whose distributions are the half-normal distribution and a power of the exponential distribution with parameter 2 respectively. The result is a distribution with greater kurtosis than the well known half-normal and slashed half-normal distributions. We studied the general density function of this distribution, with some of its properties, moments, and its coefficients of asymmetry and kurtosis. We developed the expectation–maximization algorithm and present a simulation study. We calculated the moment and maximum likelihood estimators and present three illustrations in real data sets to show the flexibility of the new model.

A Power Maxwell Distribution with Heavy Tails and Applications

Mathematics ◽

10.3390/math8071116 ◽

2020 ◽

Vol 8 (7) ◽

pp. 1116 ◽

Cited By ~ 2

Author(s):

Francisco A. Segovia ◽

Yolanda M. Gómez ◽

Osvaldo Venegas ◽

Héctor W. Gómez

Keyword(s):

Maximum Likelihood ◽

Simulation Study ◽

Heavy Tails ◽

Real Data ◽

Independent Random Variables ◽

Maxwell Distribution ◽

Moments Estimators ◽

New Distribution

In this paper we introduce a distribution which is an extension of the power Maxwell distribution. This new distribution is constructed based on the quotient of two independent random variables, the distributions of which are the power Maxwell distribution and a function of the uniform distribution (0,1) respectively. Thus the result is a distribution with greater kurtosis than the power Maxwell. We study the general density of this distribution, and some properties, moments, asymmetry and kurtosis coefficients. Maximum likelihood and moments estimators are studied. We also develop the expectation–maximization algorithm to make a simulation study and present two applications to real data.