Abstract
For many years actuaries and demographers have been doing curve fitting of age-specific mortality data. We use the eight-parameter Heligman Pollard (HP) empirical law to fit the mortality curve. It consists of three nonlinear curves, child mortality, mid-life mortality and adult mortality. It is now well-known that the eight unknown parameters in the HP law are difficult to estimate because numerical algorithms generally do not converge when model fitting is done. We consider a novel idea to fit the three curves (nonlinear splines) separately, and then connect them smoothly at the two knots. To connect the curves smoothly, we express uncertainty about the knots because these curves do not have turning points. We have important prior information about the location of the knots, and this helps in the es timation convergence problem. Thus, the Bayesian paradigm is particularly attractive. We show the theory, method and application of our approach. We discuss estimation of the curve for English and Welsh mortality data. We also make comparisons with the recent Bayesian method.
Highlights
Smoothing mortality curves is useful to actuaries and demographers for many demographic problems
Our procedure fits the three curves separately, and connect them smoothly at the two knots using a switching non-linear regression model
This paper develops a Bayesian methodology for fitting of the HP mortality curve
Summary
Smoothing mortality curves (age-specific mortality rates) is useful to actuaries and demographers for many demographic problems. We use the eight-parameter Heligman-Pollard (HP) empirical law to model mortality data across all ages. Dellaportas, Smith and Stavropoulos (2001) adopted a Bayesian approach to overcome the problems associated with the method of non-linear least squares They reported that with the use of informative priors the issue of over-parameterization is usually resolved; see Sharrow and Clark (2010). Our procedure fits the three curves (nonlinear splines) separately, and connect them smoothly at the two knots using a switching non-linear regression model. These parameters are the knots where the three curves are connected; a priori we have information about the location of these knots This motivates our Bayesian approach using the switching nonlinear regression model.
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