Insurance: Mathematics and Economics 50 (2012) 266–279 Contents lists available at SciVerse ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime Bayesian modelling of the time delay between diagnosis and settlement for Critical Illness Insurance using a Burr generalised-linear-type model Erengul Ozkok a,b,∗ , George Streftaris a , Howard R. Waters a , A. David Wilkie a a School of Mathematical and Computer Sciences, The Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK b Department of Actuarial Sciences, Hacettepe University, 06800, Beytepe, Ankara, Turkey article info Article history: Received December 2010 Received in revised form June 2011 Accepted 6 December 2011 Keywords: Bayesian analysis Burr distribution Critical illness insurance Diagnosis–settlement time lag Generalised-linear-type models Gibbs variable selection MCMC abstract We discuss Bayesian modelling of the delay between dates of diagnosis and settlement of claims in Critical Illness Insurance using a Burr distribution. The data are supplied by the UK Continuous Mortality Investigation and relate to claims settled in the years 1999–2005. There are non-recorded dates of diagnosis and settlement and these are included in the analysis as missing values using their posterior predictive distribution and MCMC methodology. The possible factors affecting the delay (age, sex, smoker status, policy type, benefit amount, etc.) are investigated under a Bayesian approach. A 3-parameter Burr generalised-linear-type model is fitted, where the covariates are linked to the mean of the distribution. Variable selection using Bayesian methodology to obtain the best model with different prior distribution setups for the parameters is also applied. In particular, Gibbs variable selection methods are considered, and results are confirmed using exact marginal likelihood findings and related Laplace approximations. For comparison purposes, a lognormal model is also considered. © 2011 Elsevier B.V. All rights reserved. 1. Introduction In this paper we model the time delay between diagnosis and settlement of claims arising in Critical Illness Insurance (CII) policies in the UK, using a Bayesian Burr model and Markov Chain Monte Carlo (MCMC) methodology. We use data supplied by the Continuous Mortality Investigation (CMI) relating to 19,127 CII claims settled in the UK in the years 1999–2005. CII is a form of long term insurance where the insurer pays a sum insured on the diagnosis during the term of one of a specified list of illnesses. Typically, regular premiums are payable throughout the term while the policy is in force. CII has been very popular in the UK, with over one million new policies issued in 2002 (CMI, 2010). Modelling the delay between diagnosis and settlement is important for several reasons. First of all, of the 19,127 claims in our data set, only 15,860 had a record of both the date of diagnosis and the date of settlement. Of the remaining 3267 claims, the vast majority had a date of settlement but not a date of diagnosis. It is important to classify the claims by date of, and hence age and ∗ Corresponding author at: Department of Actuarial Sciences, Hacettepe Univer- sity, 06800, Beytepe, Ankara, Turkey. Tel.: +90 3122976160; fax: +90 3122977998. E-mail addresses: eozkok@hacettepe.edu.tr, erengulozkok@gmail.com (E. Ozkok), G.Streftaris@hw.ac.uk (G. Streftaris), H.R.Waters@hw.ac.uk (H.R. Waters), david.wilkie@inqa.com (A.D. Wilkie). policy duration at, diagnosis, since diagnosis of a critical illness is the insured event. Where the date of settlement is known but the date of diagnosis is missing, the latter can be estimated using the date of settlement and some quantile, for example the median, of the appropriate distribution of the delay between the two events. See Ozkok (2011) and CMI (2008). Second, a model for the delay between diagnosis and settlement is important for reserving. When an illness is diagnosed, but not yet reported, there is a requirement for an IBNR reserve; when the illness is reported, but the claim has not yet been settled, there is a requirement for an IBNS reserve. Finally, the delays in settling claims beyond the time of the insured event could be taken into account in pricing. We consider a 3-parameter Burr distribution, which has many applications in actuarial science due to its high flexibility in mod- elling heavy tails. As the Burr is not a member of the exponential family of distributions, a generalised-linear-type model (GL-type) is fitted, where claim-related factors possibly affecting the delay are linked to the mean of the distribution. Beirlant et al. (1998) extended the Burr distribution to a regression model by allowing either one of its shape parameters or the scale parameter to vary with the covariates. In their paper, they estimated the parameters by using a maximum likelihood approach. Following this, Beirlant and Guillou (2001) considered extreme value methods for Pareto- type distributions under censoring, and discussed maximum likeli- hood estimates of the extreme value index of the Burr distribution, which is defined in terms of shape and scale parameters. Explana- tory variables were also regressed on this index by Beirlant and Goegebeur (2003) and maximum likelihood estimates were given. 0167-6687/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2011.12.001