Smoothing constrained generalized linear models with an application to the Lee-Carter model

Abstract

We consider a generalized linear model (GLM) with canonical link function in which parameters can be subject to (i) a set of linear constraints and (ii) smoothing. We apply Lagrange methods to give a general Newton-Raphson algorithm for such a GLM in which parameters are estimated, constraints are applied and smoothing is performed simultaneously. We express the Lee-Carter model, an important model for the forecasting of human mortality, in terms of GLMs, and use our method to estimate the parameters in the model. The smoothing option allows us to improve the forecasting properties of the model. We compare the performance of (i) the Poisson model with log link for the force of mortality and (ii) the binomial model with logit link for the probability of death in a calendar year. Examples using UK Office for National Statistics data are provided.