Bivariate discrete beta Kernel graduation of mortality data


Various parametric/nonparametric techniques have been proposed in literature to graduate mortality data as a function of age. Nonparametric approaches, as for example kernel smoothing regression, are often preferred because they do not assume any particular mortality law. Among the existing kernel smoothing approaches, the recently proposed (univariate) discrete beta kernel smoother has been shown to provide some benefits. Bivariate graduation, over age and calendar years or durations, is common practice in demography and actuarial sciences. In this paper, we generalize the discrete beta kernel smoother to the bivariate case, and we introduce an adaptive bandwidth variant that may provide additional benefits when data on exposures to the risk of death are available; furthermore, we outline a cross-validation procedure for bandwidths selection. Using simulations studies, we compare the bivariate approach proposed here with its corresponding univariate formulation and with two popular nonparametric bivariate graduation techniques, based on Epanechnikov kernels and on \(P\)-splines. To make simulations realistic, a bivariate dataset, based on probabilities of dying recorded for the US males, is used. Simulations have confirmed the gain in performance of the new bivariate approach with respect to both the univariate and the bivariate competitors.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. Bagnato L, Punzo A (2013) Finite mixtures of unimodal beta and gamma densities and the \(k\)-bumps algorithm. Comput Stat 28(4):1571–1597

    MATH  MathSciNet  Article  Google Scholar 

  2. Bagnato L, De Capitani L, Mazza A, Punzo A (2014a) SDD: serial dependence diagrams. \({\sf R}\) package version 1.1, Accessed 27 Feb 2014

  3. Bagnato L, De Capitani L, Punzo A (2014b) Testing serial independence via density-based measures of divergence. Methodol Comput Appl Prob 16(3):1–15 doi:10.1007/s11009-013-9320-4

  4. Bagnato L, De Capitani L, Mazza A, Punzo A (in press) SDD: An \({\sf R}\) package for serial dependence diagrams. J Stat Softw

  5. Bloomfield DSF, Haberman S (1987) Graduation: some experiments with kernel methods. J Inst Actuar 114(2):339–369

    Article  Google Scholar 

  6. Camarda CG (2012) Mortality smooth: an \({\sf R}\) package for smoothing Poisson counts with \(P\)-splines. J Stat Softw 50(1):1–24

    MathSciNet  Google Scholar 

  7. Chen SX (2000) Beta kernel smoothers for regression curves. Stat Sin 10(1):73–91

    MATH  Google Scholar 

  8. Copas JB, Haberman S (1983) Non-parametric graduation using kernel methods. J Inst Actuar 110(1):135–156

    Article  Google Scholar 

  9. Currie ID, Durban M, Eilers PHC (2004) Smoothing and forecasting mortality rates. Stat Modell 4(4):279–298

    MATH  MathSciNet  Article  Google Scholar 

  10. Debón A, Montes F, Sala R (2005) A comparison of parametric models for mortality graduation. Application to mortality data for the Valencia region (Spain). Stat Operat Res Trans 29(2):269–288

    MATH  Google Scholar 

  11. Debón A, Montes F, Sala R (2006a) A comparison of models for dynamic life tables. Application to mortality data from the Valencia Region (Spain). Lifetime Data Anal 12(2):223–244

    MATH  MathSciNet  Article  Google Scholar 

  12. Debón A, Montes F, Sala R (2006b) A comparison of nonparametric methods in the graduation of mortality: application to data from the Valencia Region (Spain). Int Stat Rev 74(2):215–233

    Article  Google Scholar 

  13. Elzhov TV, Mullen KM, Spiess AN, Bolker B (2013) Minpack.lm: R interface to the Levenberg-Marquardt nonlinear least-squares algorithm found in MINPACK, plus support for bounds. R package version 1.1–8, Accessed 31 Aug 2013

  14. Felipe A, Guillen M, Nielsen JP (2001) Longevity studies based on kernel hazard estimation. Insur: Math Econ 28(2):191–204

    MATH  Google Scholar 

  15. Fledelius P, Guillen M, Nielsen J, Petersen KS (2004) A comparative study of parametric and nonparametric estimators of old-age mortality in sweden. J Actuar Pract 1:101–126

    Google Scholar 

  16. Forfar DO, McCutcheon JJ, Wilkie AD (1988) On graduation by mathematical formula. J Inst Actuar 115(1):1–149

    Article  Google Scholar 

  17. Fusaro RE, Nielsen JP, Scheike TH (1993) Marker-dependent hazard estimation: an application to AIDS. Stat Med 12(9):843–865

    Article  Google Scholar 

  18. Gavin JB, Haberman S, Verrall RJ (1993) Moving weighted average graduation using kernel estimation. Insurance 12(2):113–126

    MATH  MathSciNet  Google Scholar 

  19. Gavin JB, Haberman S, Verrall RJ (1994) On the choice of bandwidth for kernel graduation. J Inst Actuar 121(1):119–134

    Article  Google Scholar 

  20. Gavin JB, Haberman S, Verrall RJ (1995) Graduation by Kernel and adaptive Kernel methods with a boundary correction. Trans Soc Actuar 47:173–209

    Google Scholar 

  21. Guillén M, Nielsen JP, Pérez-Marín AM (2006) Multiplicative hazard models for studying the evolution of mortality. Ann Actuar Sci 1(1):165–177

    Article  Google Scholar 

  22. Gupta A, Orozco-Castaeda JM, Nagar D (2011) Non-central bivariate beta distribution. Stat Pap 52(1):139–152

    MATH  Article  Google Scholar 

  23. Härdle W (1990) Applied nonparametric regression, econometric society monographs, vol 19. Cambridge University Press, Cambridge

    Book  Google Scholar 

  24. Heligman L, Pollard JH (1980) The age pattern of mortality. J Inst Actuar 107(1):49–80

    Article  Google Scholar 

  25. Human Mortality Database (2013) University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). or Accessed 11 May 2013

  26. London D (1985) Graduation: the revision of estimates. Actex Publications Abington, Connecticut

    Google Scholar 

  27. Mazza A, Punzo A (2011) Discrete beta kernel graduation of age-specific demographic indicators. In: Ingrassia S, Rocci R, Vichi M (eds) New perspectives in statistical modeling and data analysis. Studies in classification, data analysis and knowledge organization, Springer, Berlin, p 127–134

  28. Mazza A, Punzo A (2013a) Graduation by adaptive discrete beta kernels. In: Giusti A, Ritter G, Vichi M (eds) Classification and data mining. Studies in classification, data analysis and knowledge organization, Springer, Berlin, p 243–250

  29. Mazza A, Punzo A (2013b) Using the variation coefficient for adaptive discrete beta kernel graduation. In: Giudici P, Ingrassia S, Vichi M (eds) Statistical models for data analysis. Studies in classification, data analysis and knowledge organization, Springer International Publishing, Berlin, p 225–232

  30. Mazza A, Punzo A (2014a) DBKGrad: an \({\sf R}\) package for mortality rates graduation by discrete beta kernel techniques. J Stat Softw 57(Code Snippet 2):1–18

  31. Mazza A, Punzo A (2014b) DBKGrad: discrete beta kernel graduation of mortality data. \({\sf R}\) package version 1.5, Accessed 8 Apr 2014

  32. Moré J (1978) The Levenberg-Marquardt algorithm: implementation and theory. In: Watson GA (ed) Numerical analysis. Lecture notes in mathematics, vol 630, Springer, Berlin, p 104–116

  33. Nielsen JP, Linton OB (1995) Kernel estimation in a nonparametric marker dependent hazard model. Ann Stat 23(5):1735–1748

    MATH  MathSciNet  Article  Google Scholar 

  34. Olkin I, Liu R (2003) A bivariate beta distribution. Stat Prob Lett 62(4):407–412

    MATH  MathSciNet  Article  Google Scholar 

  35. Opsomer JD, Francisco-Fernández M (2010) Finding local departures from a parametric model using nonparametric regression. Stat Pap 51(1):69–84

    MATH  Article  Google Scholar 

  36. Peristera P, Kostaki A (2005) An Evaluation of the Performance of Kernel Estimators for Graduating Mortality Data. J Popul Res 22(2):185–197

    Article  Google Scholar 

  37. Punzo A (2010) Discrete beta-type models. In: Locarek-Junge H, Weihs C (eds) Classification as a tool for research. Studies in classification, data analysis and knowledge organization, Springer, Berlin, p 253–261

  38. Punzo A, Zini A (2012) Discrete approximations of continuous and mixed measures on a compact interval. Stat Pap 53(3):563–575

    MATH  MathSciNet  Article  Google Scholar 

  39. R Core Team (2013) \({\sf R}\): a language and environment for statistical computing. \({\sf R}\) Foundation for statistical computing, Vienna, Austria, Accessed 11 May 2013

  40. Richards SJ, Kirkby JG, Currie ID (2006) The importance of year of birth in two-dimensional mortality data. Br Actuar J 12(1):5–61

    Article  Google Scholar 

  41. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464

    MATH  Article  Google Scholar 

  42. Scott DW (1992) Multivariate density estimation: theory, practice, and visualization. Wiley, New York

    MATH  Book  Google Scholar 

  43. Verrall RJ (1993) Graduation by dynamic regression methods. J Inst Actuar 120(1):153–170

    Article  Google Scholar 

  44. Yi Z, Vaupel JW (2003) Oldest-old mortality in China. Demogr Res 8(7):215–244

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Antonio Punzo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mazza, A., Punzo, A. Bivariate discrete beta Kernel graduation of mortality data. Lifetime Data Anal 21, 419–433 (2015).

Download citation


  • Bivariate graduation
  • Kernel smoothing
  • Beta distribution
  • Cross-validation

Mathematics Subject Classification

  • 62G08
  • 62P05