Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T13:11:57.939Z Has data issue: false hasContentIssue false
  • English
  • Français

An Actuarial Survey of Statistical Models for Decrement and Transition Data - I: Multiple State, Poisson and Binomial Models

Published online by Cambridge University Press:  10 June 2011

A.S. Macdonald
A.S. Macdonald
Affiliation:
Department of Actuarial Maths & Stats, Heriot-Watt University, Edinburgh, EH14 4AS, U.K. Tel: +44 (0)131 451 3202; Fax: +44 (0)131 451 3249; E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper surveys some statistical models of survival data. A basic model of a random lifetime is defined, and censoring is introduced. Methods based on observations of small segments of lifetimes are compared. Markov and semi-Markov (multiple state) models are recommended as well-understood and flexible models well suited to actuarial data. A Poisson model is discussed as an approximation to a two state model, while traditional Binomial-type models are shown to be more restricted and less tractable than multiple state models.

Keywords

Actuarial EstimateBalducci AssumptionCensoringMarkov ModelsMultiple State ModelsSemi-Markov Models
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 2 , Issue 1 , 01 April 1996 , pp. 129 - 155
Copyright
Copyright © Institute and Faculty of Actuaries 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aalen, O-O. (1978). Non-parametric inference for a family of counting processes. The Annals of Statistics, 6, 701–726.CrossRefGoogle Scholar
Aalen, O.O. (1987). Dynamic modelling and causality. Scandinavian Actuarial Journal, 1987, 177–190.CrossRefGoogle Scholar
Andersen, P.K. & Borgan, O. (1985). Counting process models for life history data: a review. Scandinavian Journal of Statistics, 12, 97–158.Google Scholar
Andersen, P.K., Borgan, O., Gill, R.D. & Keiding, N. (1993). Statistical models based on counting processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Andersen, P.K. & Gill, R.D. (1982). Cox's regression model for counting processes: a large sample study. The Annals of Statistics, 10, 1100–1120.CrossRefGoogle Scholar
Arnold, B.C. & Brockett, P.L. (1983). Identifiability for dependent multiple decrement/competing risks models. Scandinavian Actuarial Journal, 1983, 117–127.CrossRefGoogle Scholar
Batten, R.W. (1978). Mortality table construction. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
Bailey, W.G. & Haycocks, H.W. (1946). Some theoretical aspects of multiple decrement tables. T. and A. Constable Ltd.Google Scholar
Bailey, W.G. & Haycocks, H.W. (1947). A synthesis of methods of deriving measures of decrement from observed data (with discussion). J.I.A. 73, 179–212.Google Scholar
Benjamin, B. & Pollard, J.H. (1980). The analysis of mortality and other actuarial statistics. Heinemann, London.Google Scholar
Borgan, ø. (1984). Maximum likelihood estimation in parametric counting process models, with applications to censored failure time data. Scandinavian Journal of Statistics, 11, 1–16.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. & Nesbitt, C.J. (1986). Actuarial mathematics. Society of Actuaries, Itasca, Il.Google Scholar
Breslow, N.E. (1974). Covariance analysis of censored survival data. Biometrics, 30, 89–100.CrossRefGoogle ScholarPubMed
Breslow, N.E. & Crowley, J. (1974). A large sample study of the life table and product limit estimates under random censorship. The Annals of Statistics, 2, 437–453.CrossRefGoogle Scholar
Breslow, N.E. (1993). Introduction to Kaplan & Meier (1958) ‘Nonparametric estimation from incomplete observations’, in Kotz, S. & Johnson, N.L. (1993). Breakthroughs in statistics II: methodology and distribution. Springer-Verlag, New York, 319–337.Google Scholar
Broffit, J.D. (1984). Maximum likelihood alternatives to actuarial estimators of mortality rates (with discussion). Transactions of the Society of Actuaries, XXXVI, 77–142.Google Scholar
Carrière, J.F. (1994). Dependent decrement theory. To appear in Transactions of the Society of Actuaries, XLVI.Google Scholar
Clarke, R.D. (1978). Mortality of impaired lives (with discussion). J.I.A. 105, 15–46.Google Scholar
Clayton, D. (1988). The analysis of event history data: a review of progress and outstanding problems. Statistics in Medicine, 7, 819–841.CrossRefGoogle Scholar
Collett, D. (1994). Modelling survival data in medical research. Chapman & Hall, London.CrossRefGoogle Scholar
Continuous Mortality Investigation Bureau (CMIB) (1988). CMIR 9.Google Scholar
Continuous Mortality Investigation Bureau (CMIB) (1993). Calculation of continuation tables and allowance for non-recorded claims based on the PHI experience 1975–78. CMIR 13, 123–130.Google Scholar
Cox, D.R. (1972). Regression models and life-tables (with discussion). J.R.S.S. B, 34, 187–220.Google Scholar
Cox, D.R. (1975). Partial likelihood. Biometrika, 62, 269–276.CrossRefGoogle Scholar
Cox, D.R. & Hinkley, D.V. (1974). Theoretical statistics. Chapman & Hall, London.CrossRefGoogle Scholar
Cox, D.R. & Oakes, D. (1984). Analysis of survival data. Chapman & Hall, London.Google Scholar
Crowder, M.J. (1991). On the identifiability crisis in competing risks analysis. Scandinavian Journal of Statistics, 18, 223–233.Google Scholar
Crowder, M. (1994). Identifiability crises in competing risks. International Statistical Review, 62, 379–391.CrossRefGoogle Scholar
Daykin, C.D., Clark, P.N.S., Eves, M.J., Haberman, S., Le Grys, D.J., Lockyer, J., Michaelson, R.W. & Wilkie, A.D. (1988). The impact of HIV infection and AIDS on insurance in the United Kingdom. J.I.A. 115, 727–838.Google Scholar
David, H.A. & Moeschberger, M.L. (1978). The theory of competing risks. Griffin, London.Google Scholar
Dorrington, R.E. & Slawski, J.K. (1993). A defence of the conventional actuarial approach to the estimation of the exposed-to-risk. Scandinavian Actuarial Journal, 1993, 107–113.Google Scholar
Elandt-Johnson, R.C. & Johnson, N.L. (1980). Survival models and data analysis. John Wiley, New York.Google Scholar
England, P.D. (1993). Statistical modelling of excess mortality of medically impaired insured lives. Ph.D. thesis, City University, London.Google Scholar
Fleming, T.R. & Harrington, D.P. (1991). Counting processes and survival analysis. John Wiley, New York.Google Scholar
Forfar, D.O., Mccutcheon, J.J. & Wilkie, A.D. (1988). On graduation by mathematical formula (with discussion). J.I.A. 115, 1–149 and 693–698, and T.F.A. 41, 97–269.Google Scholar
Gail, M. (1975). A review and critique of some models used in competing risk analysis. Biometrics, 31, 209–222.CrossRefGoogle ScholarPubMed
Gehan, E.A. (1965). A generalised Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52, 203–223.CrossRefGoogle Scholar
Gerber, H.U. (1990). Life insurance mathematics (English edition). Springer-Verlag, Berlin, and the Swiss Association of Actuaries, Zurich.CrossRefGoogle Scholar
Gill, R.D. (1980). Censoring and stochastic integrals. Mathematical Centre Tracts, 124, Mathematisch Centrum, Amsterdam.Google Scholar
Gill, R.D. (1984). Understanding Cox's regression model: a martingale approach. Journal of the American Statistical Association, 79, 441–447.CrossRefGoogle Scholar
Gill, R.D. & Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. The Annals of Statistics, 18, 1501–1555.CrossRefGoogle Scholar
Greenwood, M. (1926). The errors of sampling of the survivorship tables. Reports on Public Health and Statistical Subjects, 33, Appendix 1. H.M.S.O., London.Google Scholar
Haycocks, H.W. & Perks, W. (1955). Mortality and other investigations, Vol. I. Cambridge University Press.Google Scholar
Heckman, J.J. & Honore, B.E. (1989). The identifiability of the competing risks model. Biometrika, 76, 325–330.CrossRefGoogle Scholar
Hoem, J.M. (1969). Markov chain models in life insurance. Blätter der Deutschen Gesellschaft für Versicherungsmathematik, 9, 91–107.Google Scholar
Hoem, J.M. (1976). The statistical theory of demographic rates (with discussion). Scandinavian Journal of Statistics, 3, 169–185.Google Scholar
Hoem, J.M. (1984). A flaw in actuarial exposed-to-risk theory. Scandinavian Actuarial Journal, 1984, 187–194.CrossRefGoogle Scholar
Hoem, J.M. (1987). Statistical analysis of a multiplicative model and its application to the standardization of vital rates: a review. International Statistical Review, 55, 119–152.CrossRefGoogle Scholar
Hoem, J.M. (1988). The versatility of the Markov chain as a tool in the mathematics of life insurance. Transactions of the 23rd International Congress of Actuaries, Helsinki S, 171–202.Google Scholar
Hoem, J.M. & Funck-Jensen, U. (1982). Multistate life table methodology: a probabilist critique. In Multidimensional mathematical demography, eds. Land, K.C. & Rogers, A., 155–264, Academic Press.CrossRefGoogle Scholar
Hogg, R. & Klugman, S. (1984). Loss distributions. John Wiley, New York.CrossRefGoogle Scholar
Jacobsen, M. (1982). Statistical analysis of counting processes. Lecture Notes in Statistics, 12, Springer-Verlag, New York.Google Scholar
Jewell, W.S. (1980). Generalized models of the insurance business (life and/or non-life insurance). Transactions of the 21st International Congress of Actuaries, Zurich & Lausanne, S, 87–141.Google Scholar
Jones, B.L. (1994). Actuarial calculations using a Markov model. To appear in Transactions of the Society of Actuaries, XLVI.Google Scholar
Kalbfleisch, J.D. & Prentice, R.L. (1973). Marginal likelihoods based on Cox's regression and life model. Biometrika, 60, 267–278.CrossRefGoogle Scholar
Kalbfleisch, J.D. & Prentice, R.L. (1980). The statistical analysis of failure time data. John Wiley, New York.Google Scholar
Kaplan, E.L. & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association 53, 457—481. Reprinted in Kotz S. & Johnson N.L. (1993). Breakthroughs in statistics II: methodology and distribution. Springer-Verlag, New York, 319–337.CrossRefGoogle Scholar
Karr, A.F. (1991). Point processes and their statistical inference (2nd edition). Marcel Dekker, New York.Google Scholar
Kulkarni, V.G. (1995). Modeling and analysis of stochastic systems. Chapman & Hall, London.Google Scholar
Lancaster, T. (1990). The econometric analysis of transition data. Cambridge University Press.CrossRefGoogle Scholar
Lindsey, J.C. & Ryan, L.M. (1993). A three-state multiplicative model for rodent tumorigenicity experiments. Applied Statistics, 42, 283–300.CrossRefGoogle Scholar
Makeham, W.M. (1874). On an application of the theory of the composition of décrémentai forces. J.I.A. 18, 317–322.Google Scholar
Marshall, A.W. & Olkin, I. (1967). A bivariate exponential distribution. Journal of the American Statistical Association, 62, 30–44.CrossRefGoogle Scholar
Neill, A. (1977). Life contingencies. Heinemann, London.Google Scholar
Papatryandafylou, A. & Waters, H.R. (1984). Martingales in life insurance. Scandinavian Actuarial Journal, 1984, 210–230.CrossRefGoogle Scholar
Peto, R., Pike, M.C., Armitage, P., Breslow, N.E., Cox, D.R., Howard, S.V., Mantel, N., McPherson, K., Peto, J. & Smith, P.O. (1977). Design and analysis of randomized clinical trials requiring prolonged observation of each patient. II. Analysis and examples. British Journal of Cancer, 35, 51–67.CrossRefGoogle ScholarPubMed
Prentice, R.L., Kalbfleisch, J.D., Peterson, A.V. Jr, Flournoy, N.S., Farewell, V.T. & Breslow, N.E. (1978). The analysis of failure times in the presence of competing risks. Biometrics, 34, 541–554.CrossRefGoogle ScholarPubMed
Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions. The Annals of Statistics, II, 453–466.Google Scholar
Ramsay, C.M. (1989). AIDS and the calculation of life insurance functions (with discussion). Transactions of the Society of Actuaries, XLI, 393–422.Google Scholar
Renshaw, A.E. (1988). Modelling excess mortality using GLIM. J.I.A. 115, 299–315.Google Scholar
Schou, G. & Væth, M. (1980). A small sample study of occurrence/exposure rates for rare events. Scandinavian Actuarial Journal, 1980, 209–205.CrossRefGoogle Scholar
Scott, W.F. (1982). Some applications of the Poisson distribution in mortality studies. T.F.A. 38, 255–263.Google Scholar
Seal, H.L. (1977). Studies in the history of probability and statistics. XXXV Multiple decrements or competing risks. Biometrika, 64 429–439.Google Scholar
Smith, A.D. (1991). The use of martingales in actuarial work. Transactions of the 2nd A.F.I.R. International Colloquium, Brighton, 4, 39–81.Google Scholar
Sprague, T.B. (1879). On the construction of a combined marriage and mortality table from observations made as to the rates of marriage and mortality among any body of men. J.I.A. 21, 406–452.Google Scholar
Sverdrup, E. (1965). Estimates and test procedures in connection with stochastic models for deaths, recoveries and transfers between states of health. Skandinavisk Aktuaritidskrift, 48, 184–211.Google Scholar
Tsiatis, A.A. (1975). A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of Sciences, U.S.A., 72, 20–22.CrossRefGoogle ScholarPubMed
Waters, H.R. (1984). An approach to the study of multiple state models. J.I.A. 111, 363–374.Google Scholar
Waters, H.R. (1991a). A multiple state model for permanent health insurance. Continuous Mortality Investigation Bureau Report, 12, 5–20.Google Scholar
Waters, H.R. (1991b). Computational procedures for the model. Continuous Mortality Investigation Bureau Report, 12, 79–96.Google Scholar
Waters, H.R. & Wilkie, A.D. (1987). A short note on the construction of life tables and multiple decrement tables. J.I.A. 114, 569–580.Google Scholar
Whitehead, J. (1980). Fitting Cox's regression model to survival data using GLIM. Applied Statistics, 29, 268–275.CrossRefGoogle Scholar
Wilkie, A.D. (1988a). An actuarial model for AIDS. J.I.A. 115, 839–853.Google Scholar
Wilkie, A.D. (1988b). Markov models for combined marriage and mortality tables. Transactions of the 23rd International Congress of Actuaries, Helsinki, 3, 473–486.Google Scholar
Wood, J.W., Holman, D.J., Yashin, A.I., Peterson, R.J., Weinstein, M. & Chang, M.-C. (1994). A multistate model of fecundability and sterility. Demography, 31, 403–426.CrossRefGoogle ScholarPubMed