MODELLING INSURANCE LOSSES USING CONTAMINATED GENERALISED BETA TYPE-II DISTRIBUTION

Abstract The four-parameter distribution family, the generalised beta type-II (GB2), also known as the transformed beta distribution, has been proposed for modelling insurance losses. As special cases, this family nests many distributions with light and heavy tails, including the lognormal, gamma, Weibull, Burr and generalised gamma distributions. This paper extends the GB2 family to the contaminated GB2 family, which offers many flexible features, including bimodality and a wide range of skewness and kurtosis. Properties of the contaminated distribution are derived and evaluated in a simulation study and the suitability of the contaminated GB2 distribution for actuarial purposes is demonstrated through two real loss data sets. Analysis of tail quantiles for the data suggests large differences in extreme quantile estimates for different loss distribution assumptions, showing that the selection of appropriate distributions has a significant impact for insurance companies.

[1]  S. Nadarajah,et al.  New Folded Models for the Log-Transformed Norwegian Fire Claim Data , 2015 .

[2]  G. Peters,et al.  RISK MARGIN QUANTILE FUNCTION VIA PARAMETRIC AND NON-PARAMETRIC BAYESIAN APPROACHES , 2015, ASTIN Bulletin.

[3]  Saralees Nadarajah,et al.  New composite models for the Danish fire insurance data , 2014 .

[4]  Alice X.D. Dong,et al.  Bayesian analysis of loss reserving using dynamic models with generalized beta distribution , 2013 .

[5]  Martin Eling,et al.  Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models? , 2012 .

[6]  Chenchen Sun,et al.  Modeling with Weibull-Pareto Models , 2012 .

[7]  Edward W. Frees,et al.  A generalized beta copula with applications in modeling multivariate long-tailed data , 2011 .

[8]  A. Kleefeld,et al.  Folded and log-folded-t distributions as models for insurance loss data , 2009 .

[9]  The Weibull–Pareto Composite Family with Applications to the Analysis of Unimodal Failure Rate Data , 2009 .

[10]  Peng Shi,et al.  Actuarial Applications of a Hierarchical Insurance Claims Model , 2009, ASTIN Bulletin.

[11]  Emiliano A. Valdez,et al.  Hierarchical Insurance Claims Modeling , 2008 .

[12]  Robust Bayesian Analysis of Loss Reserves Data Using the Generalized-t Distribution , 2008, ASTIN Bulletin.

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

[14]  Richard Verrall,et al.  Predictive Distributions of Outstanding Liabilities in General Insurance , 2006, Annals of Actuarial Science.

[15]  J. Berger.  The case for objective Bayesian analysis , 2006 .

[16]  Malwane M. A. Ananda,et al.  Modeling actuarial data with a composite lognormal-Pareto model , 2005 .

[17]  Emiliano A. Valdez,et al.  Tail Conditional Expectations for Exponential Dispersion Models , 2005, ASTIN Bulletin.

[18]  Eric R. Ziegel,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2004, Technometrics.

[19]  U. Makov,et al.  Contaminated Exponential Dispersion Loss Models , 2003 .

[20]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[21]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[22]  G. Taylor,et al.  Loss Reserving: An Actuarial Perspective , 2000 .

[23]  Phhilippe Jorion.  Value at Risk: The New Benchmark for Managing Financial Risk , 2000 .

[24]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[25]  Kenneth A. Froot.  The Financing of Catastrophe Risk , 1999 .

[26]  Anthony Y. C. Kuk,et al.  The Analysis of Methadone Clinic Data Using Marginal and Conditional Logistic Models with Mixture or Random Effects , 1998 .

[27]  Stuart A. Klugman,et al.  Loss Models: From Data to Decisions , 1998 .

[28]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[29]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[30]  James B. McDonald,et al.  Partially Adaptive Estimation of Regression Models via the Generalized T Distribution , 1988, Econometric Theory.

[31]  J. McDonald,et al.  Applications of the GB2 family of distributions in modeling insurance loss processes , 1988 .

[32]  Thomas A. Aiuppa.  Evaluation of Pearson Curves as an Approximation of the Maximum Probable Annual Aggregate Loss , 1988 .

[33]  Henrik Ramlau-Hansen.  A solvency study in non-life insurance: Part 1. Analyses of fire, windstorm, and glass claims , 1988 .

[34]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[35]  Richard J. Butler,et al.  Some Generalized Mixture Distributions with an Application to Unemployment Duration , 1987 .

[36]  A. S. Paulson,et al.  A Practical Approach to Measuring the Distribution of Total Annual Claims , 1985 .

[37]  James B. McDonald,et al.  Some Generalized Functions for the Size Distribution of Income , 1984 .

[38]  J. Cummins,et al.  A Comparative Analysis of Alternative Maximum Probable Yearly Aggregate Loss Estimators , 1978 .

[39]  J. Tukey.  A survey of sampling from contaminated distributions , 1960 .