Longitudinal Modeling of Claim Counts using Jitters Emiliano A. Valdez Longitudinal Modeling of Claim Counts using Jitters joint work with Peng Shi, Northern Illinois University Introduction Background Literature Eurandom Workshop on Actuarial and Financial Statistics Modeling Eindhoven, The Netherlands, 29-30 August 2011 Random effects models Copula models Continuous extension with jitters Some properties Empirical analysis Model specification Singapore data Inference Variable selection Estimation results Model validation Concluding remarks Selected reference Emiliano A. Valdez Department of Mathematics University of Connecticut Storrs, Connecticut, USA page 1
Longitudinal Outline Modeling of Claim Counts using Jitters Introduction 1 Emiliano A. Valdez Background Literature 2 Modeling Introduction Random effects models Background Copula models Literature Modeling Continuous extension with jitters Random effects models Some properties Copula models Continuous extension with jitters Empirical analysis 3 Some properties Empirical analysis Model specification Model specification Singapore data Singapore data Inference Inference 4 Variable selection Estimation results Variable selection Model validation Estimation results Concluding remarks Selected reference Model validation 5 Concluding remarks Selected reference 6 page 2
Longitudinal Background Modeling of Claim Counts using Jitters Emiliano A. Valdez Two-part model for pure premium calculation: decompose total claims into claim frequency (number of claims) and claim severity (amount of claim, given a claim occurs). Introduction Background Several believe that the claim frequency, or claim counts, Literature Modeling is the more important component. Random effects models Copula models Past claims experience provide invaluable insight into Continuous extension with jitters some of the policyholder risk characteristics for experience Some properties Empirical analysis rating or credibility ratemaking. Model specification Singapore data Modeling longitudinal claim counts can assist to test Inference economic hypothesis within the context of a multi-period Variable selection Estimation results contract. Model validation Concluding remarks It might be insightful to explicitly measure the association Selected reference of claim counts over time (intertemporal dependence). page 3
Longitudinal Longitudinal data Modeling of Claim Counts using Jitters Emiliano A. Valdez Assume we observe claim counts, N it , for a group of policyholders i , for i = 1 , 2 , . . . , m , in an insurance portfolio over T i years. Introduction Background For each policyholder, the observable data is a vector of Literature claim counts expressed as ( N i 1 , . . . , N iT i ) . Modeling Random effects models Copula models Data may be unbalanced: length of time T i observed may Continuous extension with jitters differ among policyholders. Some properties Empirical analysis Set of observable covariates x it useful to sub-divide the Model specification portfolio into classes of risks with homogeneous Singapore data Inference characteristics. Variable selection Estimation results Here, we present an alternative approach to modeling Model validation Concluding remarks longitudinal insurance claim counts using copulas and Selected reference compare its performance with standard and traditional count regression models. page 4
Longitudinal Literature Modeling of Claim Counts using Jitters Emiliano A. Valdez Alternative models for longitudinal counts: Random effects models: the most popular approach Marginal models with serial correlation Introduction Autoregressive and integer-valued autoregressive models Background Literature Common shock models Modeling Random effects models Useful books on count regression Copula models Continuous extension with jitters Cameron and Trivedi (1998): Regression Analysis of Count Some properties Data Empirical analysis Model specification Denuit et al. (2007): Actuarial Modelling of Claim Counts: Singapore data Risk Classification, Credibility and Bonus-Malus Systems Inference Variable selection Frees (2009): Regression Modeling with Actuarial and Estimation results Model validation Financial Applications Concluding remarks Winkelmann (2010): Econometric Analysis of Count Data Selected reference The recent survey work of Boucher, Denuit and Guillén (2010) provides for a comparison of the various models. page 5
Longitudinal Literature - continued Modeling of Claim Counts using Jitters Copula regression for multivariate discrete data: Emiliano A. Valdez Increasingly becoming popular Applications found in various disciplines: Economics: Prieger (2002), Cameron et al. (2004), Zimmer Introduction and Trivedi (2006) Background Literature Biostatistics: Song et al. (2008), Madsen and Fang (2010) Modeling Actuarial science: Purcaru and Denuit (2003), Shi and Valdez Random effects models (2011) Copula models Continuous extension with jitters Modeling longitudinal insurance claim counts: Some properties Frees and Wang (2006): model joint pdf of latent variables Empirical analysis Model specification Boucher, Denuit and Guillén (2010): model joint pmf of claim Singapore data counts Inference Variable selection Estimation results Be pre-cautious when using copulas for multivariate Model validation discrete observations: non-uniqueness of the copula, Concluding remarks vague interpretation of the nature of dependence. See Selected reference Genest and Nešlehová (2007). We adopt an approach close to Madsen and Fang (2010): joint regression analysis. page 6
Longitudinal Random effects models Modeling of Claim Counts using Jitters Emiliano A. Valdez To capture the intertemporal dependence within subjects, the most popular approach is to introduce a common random effect, say α i , to each observation. Introduction Background The joint pmf for ( N i 1 , . . . , N iT i ) can be expressed as Literature Modeling Random effects models Pr ( N i 1 = n i 1 , . . . , N iT i = n iT i ) = Copula models � ∞ Continuous extension with jitters Pr ( N i 1 = n i 1 , . . . , N iT i = n iT i | α i ) f ( α i ) d α i Some properties 0 Empirical analysis Model specification Singapore data where f ( α i ) is the density function of the random effect. Inference Variable selection Typical assumption is conditional independence as follows: Estimation results Model validation Pr ( N i 1 = n i 1 , . . . , N iT i = n iT i | α i ) = Concluding remarks Selected reference Pr ( N i 1 = n i 1 | α i ) × · · · × Pr ( N iT i = n iT i | α i ) . page 7
Longitudinal Some known random effects models Modeling of Claim Counts using Jitters Emiliano A. Valdez Poisson N it ∼ Poisson (˜ λ it ) ′ ˜ λ it = η i λ it = η i ω it exp ( x it β ) , and η i ∼ Gamma ( ψ, ψ ) ˜ ′ it β ) , and α i ∼ N ( 0 , σ 2 ) λ it = ω it exp ( α i + x Introduction Background Negative Binomial Literature Modeling NB1: 1 + 1 /ν i ∼ Beta ( a , b ) Random effects models � λ it � � n it Copula models � Γ( n it + λ it ) ν i 1 Pr ( N it = n it | ν i ) = Continuous extension with Γ( λ it )Γ( n it + 1 ) 1 + ν i 1 + ν i jitters Some properties NB2: α i ∼ N ( 0 , σ 2 ) Empirical analysis � ψ � � n it � ˜ Γ( n it + ψ ) Model specification ψ λ it Pr ( N it = n it | α i ) = ˜ ˜ Singapore data Γ( ψ )Γ( n it + 1 ) λ it + ψ λ it + ψ Inference Zero-inflated models Variable selection � π it + ( 1 − π it ) f ( n it | α i ) Estimation results if n it = 0 Model validation Pr ( N it = n it | δ i , α i ) = if n it > 0 . Concluding remarks ( 1 − π it ) f ( n it | α i ) Selected reference � � � ′ π it log � δ i = δ i + z it γ , � 1 − π it ZIP ( f ∼ Poisson) and ZINB ( f ∼ NB ) page 8
Longitudinal Copula models Modeling of Claim Counts using Jitters Emiliano A. Valdez Joint pmf using copula: Pr ( N i 1 = n i 1 , . . . , N iT = n iT ) = Introduction Background 2 2 Literature ( − 1 ) j 1 + ··· + j T C ( u 1 j 1 , . . . , u Tj T ) � � · · · Modeling Random effects models j 1 = 1 j T = 1 Copula models Continuous extension with jitters Here, u t 1 = F it ( n it ) , u t 2 = F it ( n it − 1 ) , and F it denotes Some properties the distribution of N it Empirical analysis Model specification Singapore data Downside of the above specification: Inference contains 2 T terms and becomes unmanageable for large T Variable selection Estimation results Model validation involves high-dimensional integration Concluding remarks other critiques for the case of multivariate discrete data: see Selected reference Genest and Nˇ eslehová (2007) page 9
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