State space modeling of non-standard actuarial time series. (English) Zbl 0764.62087

Summary: Insurance professionals have long recognized the value of time series methods in analyzing sequential business and economic data. However, the usual class of Box-Jenkins models are sometimes not flexible enough to adequately describe many practically-arising time series. By way of contrast, the state-space model offers a general approach for multivariate time-series modeling, forecasting, and smoothing. Versions of these models having linear mean structure and Gaussian error distributions have enjoyed acturial application (for instance, in credibility theory) using the simple recursive updating formulae provided by the Kalman filter algorithm. Recently, fitting these models in more challenging nonlinear and non-Gaussian scenarios has become feasible using Monte Carlo integration techniques.
This paper gives a brief review of this new methodology, and subsequently demonstrates its usefulness in actuarial settings via three data examples. The non-standard modeling features illustrated include the explicit incorporation of covariates, direct modeling of non-stationary series without differencing, multivariate analysis, comparison of results under normal and non-normal error assumptions, density estimation for future observations, and nonlinear model building and testing.


62P05 Applications of statistics to actuarial sciences and financial mathematics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F15 Bayesian inference


Full Text: DOI


[1] Andrews, D.F.; Mallows, C.L., Scale mixtures of normality, Journal of the royal statistical society, 36, 99-102, (1974), Ser. B · Zbl 0282.62017
[2] Berger, J.O., ()
[3] Besag, J., Spatial interaction and the statistical analysis of lattice systems (with discussion), Journal of the royal statistical society, 36, 192-236, (1974), Ser. B · Zbl 0327.60067
[4] Box, G.E.P.; Jenkins, G.M., ()
[5] Box, G.E.P.; Tiao, G.C., Bayesian inference in statistical analysis, (1973), Addison-Wesley Reading, MA · Zbl 0178.22003
[6] Carlin, B.P.; Polson, N.G., Inference for non-conjugate Bayesian models using the Gibbs sampler, Canadian journal of statistics, 19, 399-405, (1991) · Zbl 0850.62285
[7] Carlin, B.P.; Polson, N.G.; Stoffer, D.S., A Monte Carlo approach to nonnormal and nonlinear state space modeling, Journal of the American statistical association, 87, 493-500, (1992)
[8] De Jong, P.; Zehnwirth, B., Credibility theory and the Kalman filter, Insurance: mathematics and economics, 2, 281-286, (1983) · Zbl 0559.62084
[9] Devroye, L., Non-uniform random variate generation, (1986), Springer-Verlag New York · Zbl 0593.65005
[10] Fama, E.F., Efficient capital markets: A review of theory and empirical work, Journal of finance, 25, 383-417, (1970)
[11] Fama, E.F., Short-term interest rates as predictors of inflation, American economic review, 65, 269-282, (1975)
[12] Gelfand, A.E.; Hills, S.E.; Racine-Poon, A.; Smith, A.F.M., Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American statistical association, 85, 972-985, (1990)
[13] Gelfand, A.E.; Smith, A.F.M., Sampling based approaches to calculating marginal densities, Journal of the American statistical association, 85, 398-409, (1990) · Zbl 0702.62020
[14] Gelman, A.; Rubin, D.B., Inference from iterative simulation using multiple sequences, Statistical science, (1992), To appear (with discussion) · Zbl 1386.65060
[15] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE transactions on pattern analysis and machine intelligence, 6, 721-741, (1984) · Zbl 0573.62030
[16] Gilks, W.R.; Wild, P., Adaptive rejection sampling for Gibbs sampling, Journal of the royal statistical society, 41, 337-348, (1992), Series C (Applied Statistics) · Zbl 0825.62407
[17] Kalman, R.E., A new approach to linear filtering and prediction problems, Journal of basic engineering, 82, 34-45, (1960)
[18] Ledolter, J.; Klugman, S.; Lee, C.-S., Credibility models with time-varying trend components, ASTIN bulletin, 21, 73-91, (1991)
[19] Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E., Equations of state calculations by fast computing machines, Journal of chemical physics, 21, 1087-1091, (1953)
[20] Minitab reference manual, (1989), Minitab, Inc State College, PA
[21] Newbold, P.; Bos, T., Stochastic parameter regression models, (1985), Sage Publications Beverly Hills, CA, Sage University paper series on quantitative applications in the social sciences
[22] Odell, P.L.; Feiveson, A.H., A numerical procedure to generate a sample covariance matrix, Journal of the American statistical association, 61, 198-203, (1966)
[23] Schervish, M.J.; Carlin, B.P., On the convergence of successive substitution sampling, Journal of computational and graphical statistics, 1, 111-127, (1992)
[24] Tierney, L., Markov chains for exploring posterior distributions, Technical report no. 560, (1991), School of Statistics, University of Minnesota Minneapolis, MN
[25] West, M.; Harrison, P.J., Bayesian forecasting and dynamic models, (1989), Springer-Verlag New York · Zbl 0697.62029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.