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An evaluation of the performance of kernel estimators for graduating mortality data

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Abstract

In the graduation of the age-specific mortality pattern, recent emphasis has been placed on the use of kernel regression estimators. Three such estimators are the Nadaraya-Watson, Gasser-Muller and kernel weighted local linear estimators. This paper discusses the theoretical background of each estimator and evaluates their accuracy in graduating age-specific mortality using data for France, Japan and Sweden. For comparison, we also fit the Heligman-Pollard model and its nine-parameter variant by Kostaki. The Gasser-Muller estimator is found to be superior to the two other kernel estimators in that it is both more stable and not influenced by boundary effects. Furthermore, compared with the two parametrric models, the Gasser-Muller estimator provides a more satisfactory graduation, especially at older adult ages, in terms both of smoothness and of fidelity between the observed and graduated rates.

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References

  • Antoniadis, A., J. Bigot and T. Sapatinas. 2001. Wavelet estimators in nonparametric regression: a comparative simulation study.Journal of Statistical Software 6 (6): 1–83.

    Google Scholar 

  • Benjamin, B. and J.H. Pollard. 1986.The Analysis of Mortality Data and Other Actuarial Statistics. London: Heinemann.

    Google Scholar 

  • Blomfield, D.S.F. and S. Haberman. 1987. Graduation: some experiments with kernel methods,Journal of the Institute of Actuaries 457: 339–368.

    Google Scholar 

  • Booth, H. 2006. Demographic forecasting: 1980 to 2005 in review.International Journal of Forecasting 22: 547–581.

    Article  Google Scholar 

  • Bowman A.W. and A. Azzalini. 1997.Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-plus Illustrations. Oxford: Oxford Science Publications.

    Google Scholar 

  • Brass, W. 1971. On the scale of mortality. Pp. 66–110 in W. Brass (ed.),Biological Aspects of Demography. London: Taylor and Francis, Ltd.

    Google Scholar 

  • Brockmann, M., T. Gasser and E. Herrmann. 1993. Locally adaptive bandwidth choice for kernel regression estimators.Journal of the American Statistical Association Theory and Methods 88 (424): 1302–1309.

    Article  Google Scholar 

  • Chu, C.-K. and J.S. Marron. 1991. Choosing a kernel regression estimator.Statistical Science 6 (4): 404–436.

    Article  Google Scholar 

  • Cleveland, W. S. 1979. Robust locally weighted regression and smoothing scatterplots.Journal of the American Statistical Association 74: 829–836.

    Article  Google Scholar 

  • Copas, J.B. and S. Haberman. 1983. Non-parametric graduation using kernel methods.Journal of the Institute of Actuaries 110: 135–156.

    Google Scholar 

  • Currie, I.D., M. Durban and P.H.C. Eilers. 2004. Smoothing and forecasting mortality rates.Statistical Modelling 4(4): 279–298.

    Article  Google Scholar 

  • Epanechnikov, V.A. 1969. Non-parametric estimation of a multivariate probability density.Theory and Probability Applications 14: 153–158.

    Article  Google Scholar 

  • Forfar, D. O., J.J. McCutcheon and A.D. Wilkie. 1988. On graduation by mathematical formula.Journal of the Institute of Actuaries 115: 1–135.

    Google Scholar 

  • Gasser, T., A. Kneip and W. Kohler. 1991. A flexible and fast method for automatic smoothing.Journal of the American Statistical Society Theory and Methods 86(415): 643–652.

    Google Scholar 

  • Gasser, T. and H.G. Muller. 1979. Kernel estimation of regression functions.Smoothing Techniques for Curve Estimation. Lecture Notes in Mathematics 757: 23–68.

    Article  Google Scholar 

  • Gasser, T. and H.G. Muller. 1984. Estimating regression functions and their derivatives by the kernel method.Scandinavian Journal of Statistics 11: 171–185.

    Google Scholar 

  • Gavin, J.B., S. Haberman and R.J. Verrall. 1993. Moving weighted average graduation using kernel estimation.Insurance: Mathematics and Economics 12: 113–126.

    Article  Google Scholar 

  • Gavin, J.B., S. Haberman and R.J. Verrall. 1994. On the choice of bandwidth for kernel graduation.Journal of the Institute of Actuaries 121: 119–134.

    Google Scholar 

  • Gavin, J., S. Haberman and R. Verrall. 1995. Graduation by kernel and adaptive kernel methods with a boundary correction.Transactions of Society of Actuaries 47: 173–209.

    Google Scholar 

  • Gill, P. and W. Murray. 1978. Algorithms for the solution of a non-linear least squares problem.Journal of Numerical Analysis SIAM 15: 977–992.

    Article  Google Scholar 

  • Hannerz, H. 1999.Methodology and Applications of a New Law of Mortality. Lund: Department of Statistics, University of Lund.

    Google Scholar 

  • Hardle, W. 1990.Applied Non-parametric Regression. Cambridge: Cambridge University Press.

    Google Scholar 

  • Hardle, W. 1991.Smoothing Techniques with Implementation in S. New York: Springer-Verlag.

    Google Scholar 

  • Hartmann, M. 1987. Past and recent attempts to model mortality at all ages.Journal of Official Statistics 3: 19–36.

    Google Scholar 

  • Hastie, T. and C. Loader. 1993. Local regression: automatic kernel carpentry.Science 3(2): 120–143.

    Google Scholar 

  • Heligman, M. and J.H. Pollard. 1980. The age pattern of mortality.Journal of the Institute of Actuaries 107: 49–80.

    Google Scholar 

  • Hermann, E. 1997. Local bandwidth choice in kernel regression estimation.Journal of Computational and Graphical Statistics 6(1): 35–54.

    Article  Google Scholar 

  • Human Mortality Database. <http://www.mortality.org>. Date accessed 2002.

  • Hyndman, R.J. and S. Ullah. 2005. Robust forecasting of mortality and fertility rates: a functional data approach.Working Paper 2/05. Department of Econometrics and Business Statistics, Monash University, Melbourne.

    Google Scholar 

  • Jones, M.C., S.J. Davis and B.U. Park. 1994. Versions of kernel-type regression estimators.Journal of the American Statistical Association Theory and Methods 89 (427): 825–832.

    Article  Google Scholar 

  • Kostaki, A. 1992. A nine-parameter version of the Heligman-Pollard formula.Mathematical Population Studies 3 (4): 277–288.

    Article  Google Scholar 

  • Lokern library for R software. 1997. <http://www.unizh.ch/biostat/software>. Data accessed 2002.

  • Mack, Y.P. 1981. Local properties of k-NN regression estimates.SIAM Journal of Algebraic and Discrete Methods 2: 311–323.

    Article  Google Scholar 

  • Nadaraya, E.A., 1964. On estimating regression.Theory and Probability of Its Applications 9: 141–142.

    Article  Google Scholar 

  • Park, B.U. and J.S. Marron. 1990. Comparison of data-driven bandwidth selectors.Journal of the American Statistical Association Theory and Methods 85 (409): 66–72.

    Article  Google Scholar 

  • Ramlau-Hansen, H. 1983. The choice of a kernel function in the graduation of counting process intensities.Scandinavian Actuarial Journal: 165–182.

  • Rosenblatt, M. 1971. Curve estimates.Annals of Mathematical Statistics 42: 1815–1842.

    Article  Google Scholar 

  • Ruppert, D., S.J. Sheather and M.P. Wand. 1995. An effective bandwidth selector for local least squares regression.Journal of the American Statistical Association Theory and Methods 90 (432): 1257–1270.

    Article  Google Scholar 

  • Schwartz, S.C.. 1967. Estimation of a probability density by an orthogonal series.Annals of Mathematical Statistics 38:1261–1265.

    Article  Google Scholar 

  • Watson, G.S. 1964. Smooth regression analysis.Sankhya Series A 26: 359–372.

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Athens University of Economics and Business, Patission 76, Athens, Greece

    Paraskevi Peristera & Anastasia Kostaki

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  1. Paraskevi Peristera
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  2. Anastasia Kostaki
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Correspondence to Paraskevi Peristera.

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Peristera, P., Kostaki, A. An evaluation of the performance of kernel estimators for graduating mortality data. Journal of Population Research 22, 185–197 (2005). https://doi.org/10.1007/BF03031828

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