The determination of mortality rates from observed data

This thesis considers the determination of the mortality rate qx from full data for the year of age x to x+1 in which some lives enter or leave observation during the year of age. It commences with a survey of previous literature concerned with mortality rate estimators. New aspects of the conventional actuarial, method of moments, product limit and maximum likelihood mortality rate estimators are identified and discussed. In particular, a fresh rationale of the conventional actuarial estimator is developed and it is argued that Hoem (1984) is incorrect in claiming that this estimator is flawed. It is also argued that Hoem’s suggested approximated “operational moment relations” estimators are not satisfactory. The rectangular hyperbolic mortality law, a two-parameter mortality law embracing three common mortality assumptions as special cases, is developed and investigated. Its application in the estimation of qx is considered, including by means of the maximum likelihood estimation of the two parameters, a method included in the later simulations. Similar attention is also given to the Gompertz law. A general theory of mortality rate estimators is developed which generates most, if not all, established mortality rate estimators that assume parametric mortality laws, particular estimators being obtained by the choice of appropriate weighting functions applied to the elements of the year of age. It is shown that all estimators, assuming a one-parameter mortality law, obtained from the general theory are asymptotically unbiased, when the correct mortality law has been assumed. Use of the general theory also identifies the mortality law assumptions for which different mortality rate estimators coincide. A number of new mortality rate estimators are identified in the thesis, some through application of the general theory. Computer models are used to generate simulated mortality data in order to examine the performance of a wide range of mortality rate estimators, including a number developed in the thesis. Among other things, the results support the view that the conventional actuarial estimator is not flawed