Electronic Journal of Statistics

Generalised additive dependency inflated models including aggregated covariates

Young K. Lee, Enno Mammen, Jens P. Nielsen, and Byeong U. Park

Full-text: Open access


Let us assume that $X$, $Y$ and $U$ are observed and that the conditional mean of $U$ given $X$ and $Y$ can be expressed via an additive dependency of $X$, $\lambda(X)Y$ and $X+Y$ for some unspecified function $\lambda$. This structured regression model can be transferred to a hazard model or a density model when applied on some appropriate grid, and has important forecasting applications via structured marker dependent hazards models or structured density models including age-period-cohort relationships. The structured regression model is also important when the severity of the dependent variable has a complicated dependency on waiting times $X$, $Y$ and the total waiting time $X+Y$. In case the conditional mean of $U$ approximates a density, the regression model can be used to analyse the age-period-cohort model, also when exposure data are not available. In case the conditional mean of $U$ approximates a marker dependent hazard, the regression model introduces new relevant age-period-cohort time scale interdependencies in understanding longevity. A direct use of the regression relationship introduced in this paper is the estimation of the severity of outstanding liabilities in non-life insurance companies. The technical approach taken is to use B-splines to capture the underlying one-dimensional unspecified functions. It is shown via finite sample simulation studies and an application for forecasting future asbestos related deaths in the UK that the B-spline approach works well in practice. Special consideration has been given to ensure identifiability of all models considered.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 67-93.

Received: December 2017
First available in Project Euclid: 4 January 2019

Permanent link to this document

Digital Object Identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Structured nonparametric models age-period-cohort model identifiability B-splines UK mesothelioma mortality

Creative Commons Attribution 4.0 International License.


Lee, Young K.; Mammen, Enno; Nielsen, Jens P.; Park, Byeong U. Generalised additive dependency inflated models including aggregated covariates. Electron. J. Statist. 13 (2019), no. 1, 67--93. doi:10.1214/18-EJS1515. https://projecteuclid.org/euclid.ejs/1546570942

Export citation


  • Agmon, S. (1965)., Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ.
  • Antonczyk, D., Fitzenberger, B., Mammen, E. and Yu, K. (2017). A nonparametric approach to identify age, time and cohort effects., Preprint.
  • Beutner, E. A., Reese, S. B. and Urbain, J. P. (2017). Identifiability issues of age-period and age-period-cohort models of the Lee-Carter type., Insurance: Mathematics and Economics 75, 117–125.
  • Birman, M. Š., Solomjak, M. J. (1967). Piecewise polynomial approximations of functions of classes $W_p^\alpha$ (Russian)., Mat. Sb. (N.S.) 73, 331–355.
  • Cairns, A. J. G., Blake, D., Dowdb, K., Coughlan, G. D. and Epstein, D. (2011). Mortality density forecasts: An analysis of six stochastic mortality models., Insurance: Mathematics and Economics 48, 355–367.
  • Hiabu, M., Mammen, E., Martinez-Miranda, M. D. and Nielsen, J. P. (2016). In-sample forecasting with local linear survival densities., Biometrika 103, 843–859.
  • Hodgson, J. T., McElvenny, D. M. and Darnton, A. J. (2005). The expected burden of mesothelioma mortality in Great Britain from 2002 to 2050., British Journal of Cancer 92, 587–593.
  • Horowitz, J. L. and Mammen, E. (2007). Rate-optimal estimation for a general class of nonparametric regression models with unknown link function., Annals of Statistics 35, 2589–2619.
  • Kuang, D., Nielsen, B. and Nielsen, J. P. (2008a). Identification of the age-period-cohort model and the extended chain-ladder model., Biometrika 95, 979–986.
  • Kuang, D., Nielsen, B. and Nielsen, J. P. (2008b). Forecasting with the age-period-cohort model and the extended chain-ladder model., Biometrika 95, 987–991.
  • Kuang, D., Nielsen, B. and Nielsen, J. P. (2011). Forecasting in an extended chain-ladder-type model., Journal of Risk and Insurance 78, 345–359.
  • Lee, R. D. and Carter, L. R. (1992). Modeling and forecasting US mortality., Journal of American Statistical Association 87, 659–675
  • Lee, R. D. and Miller, T. (2001). Evaluating the performance of the Lee–Carter method for forecasting mortality., Demography 38, 537–549
  • Lee, Y. K., Mammen, E., Nielsen, J. P. and Park, B. U. ((2015). Asymptotics for In-Sample Density Forecasting., Annals of Statistics 43, 620–651.
  • Lee, Y. K., Mammen, E., Nielsen, J. P. and Park, B. U. (2017). Operational time and in-sample density forecasting., Annals of Statistics 45, 1312–1341.
  • Mammen, E. and Nielsen, J. P. (2003). Generalised structured models., Biometrika 90, 551–566.
  • Mammen, E., and van de Geer, S. (1997). Penalized quasi-likelihood estimation in partial linear models., Annals of Statistics 25, 1014–1035.
  • Martinez-Miranda, M. D., Nielsen, B. and Nielsen, J. P. (2015). Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality., Journal of Royal Statistical Society Series A 178, 29–55.
  • Martinez-Miranda, M. D., Nielsen, B. and Nielsen, J. P. (2016). Simple benchmark for mesothelioma projection for Great Britain., Occupational and Environmental Medicine 73, 561–563.
  • Nielsen, B. and Nielsen, J. P. (2014). Identification and forecasting in mortality models., The Scientific World Journal, 347043.
  • O-Brien, R. (2014)., Age-Period-Cohort Models: Approaches and Analyses with Aggregate Data. Chapman & Hall/CRC Press, London.
  • Peto, J., Hodgson, J. T. and Matthews, F. E. (1995). Continuing increase in mesothelioma mortality in Britain., Lancet 345, 535–539.
  • Rake, C., Gilham, C. and Hatch, J. (2009). Occupational, domestic and environmental mesothelioma risks in the British population: a case-control study., British Journal of Cancer 100, 1175–1183.
  • Renshaw, A. J. and Haberman, S. (2006). A cohort-based extension to the Lee–Carter model for mortality reduction factors., Insurance: Mathematics and Economics 38, 556–570.
  • Riebler, A., Held, L. and Rue, H. (2012). Estimation and extrapolation of time trends in registry data-Borrowing strength from related populations., Annals of Applied Statistics 6, 304–333.
  • Smith, T. R. and Wakefield, J. (2016). A review and comparison of age-period-cohort models for cancer incidence., Statistical Science 31, 591–610.
  • Tan, E., Warren, N. and Darnton, A. J. (2010). Projection of mesothelioma mortality in Britain using Bayesian methods., British Journal of Cancer 103, 430–436.
  • Tan, E., Warren, N., Darnton, A. J. (2011). Modelling mesothelioma mortality in Great Britain using the two-stage clonal expansion model., Occupational and Environmental Medicine 68, A60. doi:10.1136/oemed-2011-100382.194
  • van de Geer, S. (2000)., Empirical Processes in M-Estimation. Cambridge University Press.