Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T23:27:14.588Z Has data issue: false hasContentIssue false
  • English
  • Français

On Graduation By Mathematical Formula

Published online by Cambridge University Press:  03 October 2014

D. O. Forfar ,
J. J. McCutcheon  and
A. D. Wilkie
Get access

Extract

In the course of work undertaken as members of the Executive Committee of the Continuous Mortality Investigation Bureau in the preparation of graduated tables of mortality for the experiences of 1979–82, we have had occasion to make use of and develop a number of statistical techniques with which actuaries may not be familiar, and which are not fully discussed in the current textbook by Benjamin & Pollard (1980), though some of them have been referred to in previous papers by the CMI Committee (1974, 1976). We therefore felt that it would be useful to the profession if we were to present these methods comprehensively in one paper. We do this with the permission of the other members of the CMI Committee, who do not, however, take responsibility for what follows, whether good or bad.

Type
Research Article
Information
Transactions of the Faculty of Actuaries , Volume 41 , 1987 , pp. 97 - 269
Copyright
Copyright © Institute and Faculty of Actuaries 1987

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

Akaike, H. (1978). On the likelihood of a Time-series Model. The Statistician, 27, 217.CrossRefGoogle Scholar
Akaike, H. (1985). ‘Prediction and Entropy’ in A Celebration of Statistics, edited by Atkinson, C. & Fienberg, S. E., Springer Verlag, New York.Google Scholar
Barnett, H. A. R. (1951). Graduation Tests and Experiments, J.I.A., 77, 15.Google Scholar
Barnett, H. A. R. (1985). Criteria of Smoothness, J.I.A., 112, 331.Google Scholar
Batten, R. W. (1978). Mortality Table Construction. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
Beard, R. E. & Perks, W. (1949). The Relation between the Distribution of Sickness and the Effect of Duplicates on the Distribution of Deaths. J.I.A., 75, 75.Google Scholar
Benjamin, B. & Pollard, J.H. (1980). The Analysis of Mortality and Other Actuarial Statistics, Second Edition. Heinemann, London.Google Scholar
CMI Committee (1957). Continuous Investigation into the Mortality of Assured Lives, Memorandum on a Special Inquiry into the Distribution of Duplicate Policies. J.I.A., 83, 34 and T.F.A., 24, 94.Google Scholar
CMI Committee (1974). Considerations Affecting the Preparation of Standard Tables of Mortality, J.I.A., 101, 133 and T.F.A., 34, 135.Google Scholar
CMI Committee (1976). The Graduation of Pensioners' and of Annuitants' Mortality Experience 1967–1970, CMIR, 2, 57.Google Scholar
CMI Committee (1986). An investigation into the distribution of policies per life assured in the cause of death investigation data, CMIR, 8, 49.Google Scholar
CMI Committee (1986). The C.M.I. Bureau: A Note on the History of the Computerization of the Work of the Bureau and the Development of improved Services to Contributing Offices (Appendix: The Kolmogorov-Smirnov Test). CMIR, 8, 59.Google Scholar
Conte, S. D. & de Boor, C. (1980), Elementary Numerical Analysis, Third Edition. McGraw-Hill Kogakusha, Tokyo.Google Scholar
Cox, D. R. & Oakes, D. (1984). Analysis of Survival Data. Chapman and Hall, London.Google Scholar
Daw, R. H. (1946). On the Validity of Statistical Tests of the Graduation of a Mortality Table. J.I.A., 72, 174.Google Scholar
Daw, R. H. (1951). Duplicate Policies in Mortality Data. J.I.A., 77, 261.Google Scholar
de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag. New YorkCrossRefGoogle Scholar
Durbin, J. (1973). Distribution Theory for Tests Based on the Sample Distribution Function. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Elandt-Johnson, R. C. & Johnson, N. L. (1980). Survival Models and Data Analysis. John Wiley, New York.Google Scholar
Edwards, A. W. F. (1972). Likelihood. Cambridge University Press.Google Scholar
Fisz, M. (1963). Probability Theory and Mathematical Statistics. John Wiley. New York.Google Scholar
Forfar. D. O. & Smith, D. M. (1987). The changing shape of English Life Tables T.F.A., 40, 98.Google Scholar
Greville, T. N. E. (1978). Estimation of the Rate of Mortality in the Presence of In-and-out Movement, ARCH, 1978. 2, 41.Google Scholar
Heligman, L. & Pollard, J. H. (1980). The Age Pattern of Mortality, J.I.A., 107, 49.Google Scholar
Hoem, J. (1980). Exposed-to-risk Considerations based on the Balducci assumption and other assumptions in the analysis of mortality, ARCH, 1980. 1, 47.Google Scholar
Hoem J. (1984). A Flaw in Actuarial Exposed-to-Risk Theory. Scandinavian Actuarial Journal, 1984, 187.Google Scholar
Hogg, R. V. & Klugman, S. A. (1984). Loss Distributions. John Wiley, New York.CrossRefGoogle Scholar
Kalbfleisch, J. D. & Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. John Wiley. New York.Google Scholar
Kendall, M. G. & Stuart, A. (1979). The Advanced Theory of Statistics, Volume 2, Fourth Edition. Charles Griffin. London.Google Scholar
Larson, H. J. (1982). Introduction to Probability Theory and Statistical Inference, Third Edition. John Wiley. New York.Google Scholar
London, D. (1985). Graduation: The Revision of Estimates. Actex Publications, Winsted and Abington. Conn.Google Scholar
Mccutcheon, J. J. (1971). Some remarks on the basic mortality functions, T.F.A., 32, 395.Google Scholar
Mccutcheon, J. J. ( 1977). Some Elementary Life Table Approximations, T.F.A., 35, 297.Google Scholar
Mccutcheon, J. J. (1981). Some Remarks on Splines, T.F.A., 37, 421.Google Scholar
Mccutcheon, J. J. (1982). Graduation of the Experience of Female Assured Lives 1975–78, T.F.A., 38, 193.Google Scholar
Mccutcheon, J. J. (1983). On estimating the Force of Mortality. T.F.A., 38, 407.Google Scholar
Mccutcheon, J. J. (1984). Spline Graduation with Variable Knots, Proceedings 22nd I.C.A., 4, 47.Google Scholar
Mccutcheon, J. J. (1987). Experiments in graduating the data for the English Life Tables No. 14, T.F.A. 40, 135.Google Scholar
Mann, N. R., Schafer, R. E. & Singpurwalla, N. D. (1974). Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York.Google Scholar
Maturity Guarantees Working Party (1980). Report, J.I.A., 107, 101.Google Scholar
Nelder, J. A. & Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1986). Numerical Recipes. Cambridge University Press.Google Scholar
Rao, C. R. (1973). Linear Statistical Inference and its Applications, Second Edition. John Wiley, New York.CrossRefGoogle Scholar
Rao, S. S. (1984). Optimization Theory and Applications, Second Edition. Wiley Eastern Ltd. New Delhi.Google Scholar
Roberts, L. A. (1986). Bias in decremental rate estimates, OARD (39). Institute of Actuaries.Google Scholar
Rubinstein, R. Y. (1986). Simulation and the Monte Carlo Method. John Wiley, New York.Google Scholar
Seal, H. (1945). Tests of a mortality table graduation. J.I.A., 71, 3.Google Scholar
Scott, W. F. (1982). Some Applications of the Poisson Distribution in Mortality Studies, T.F.A., 38, 255.Google Scholar
Scott, W. F. (1986). On the calculation of the Exposed-to-Risk in the CMI investigations. Unpublished note.Google Scholar
Sverdrup, E. (1965). Estimates and Test Procedure in Connection with Stochastic Models for Deaths, Recoveries and Transfers between different States of Health, Skandinavisk Aktuarietidskrift, 48, 184.Google Scholar
Van der Waerden, B. L. (1969). Mathematical Statistics. George Allan and Unwin, London.CrossRefGoogle 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.Google Scholar

References

(1) Armitage, P. (1959) The comparison of survival curves. J. R. Stat. Soc. (A) 122: 279–300.Google Scholar
(2) Benjamin, B. (1985) Underlying theory of actuarial analysis. U. S. National Cancer Institute Managt. 67. 7–14.Google Scholar
(3) Breslow, N. E., Lubin, J. H., Marek, P. et al. ,: (1983) Multiplicative models and cohort analysis. J. Am. Stat. Assoc. 78:1–12.CrossRefGoogle Scholar
(4) Brillinger, D. R. (1961) A justification of some common laws of mortality. Society of Actuaries, Trans. 13: 116–126.Google Scholar
(5) Cox, D. R. (1972) Regression models and life table. J. R. Stat. Soc. (B) 34: 187–220.Google Scholar
(6) Kaplan, E. L., Meier, P. (1958) Non-parametric estimation from incomplete observations. J. Am. Stat. Assoc. 53: 459–481.CrossRefGoogle Scholar
(7) Littell, A. S. (1952) Estimation of the T-year survival rate from following studies over a limited period of time. Hum. Biol. 24: 87–116.Google Scholar