Graduating Mortality Rates via Divergences

One of the most important tasks in actuarial science is to describe the actual but unknown mortality pattern of a population. In order to achieve this, the actuary calculates from raw data the crude mortality rates, which usually form an irregular series. Because of this, it is common to revise the initial estimates with the aim of producing smoother estimates, with a procedure called graduation. There are several parametric and non-parametric methods to achieve this. In this paper initially we critically review the method of graduation using information theoretic ideas. Starting with Brockett’s idea to use the Kullback Leibler divergence (Brockett, 1991) we explore the use of a family of divergence indices the family of power divergence statistics (Read and Cressie, 1998) with analogous linear and/or quadratic constraints with the aim of finding the best divergence to use in order to obtain the “best” graduation. “Bestness” is defined as an acceptable level of both smoothness and goodness of fit. The results so far indicate that divergences with non-probability vectors in their arguments, as is the case with mortality rates, share, under some conditions, some of the properties of probabilistic or information theoretic divergences. The power divergence statistics also give results equivalent to those that other frequently used graduation methods give. A numerical investigation did not produce the best value or best values of , the power of the divergence statistic, for best graduation. The topic is under further investigation.