A Bayesian Adjustment of the HP Law via a Switching Nonlinear Regression Model

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.