Rieder, H. (1996): Estimation of Mortalities. Sonderforschungsbereich 386, Discussion Paper 28




If a linear regression is fit to log-transformed mortalities and the estimate is back-transformed according to the formula Ee^Y = e^{\mu + \sigma^2/2} a systematic bias occurs unless the error distribution is normal and the scale estimate is gauged to normal variance. This result is a consequence of the uniqueness theorem for the Laplace transform. We determine the systematic bias of minimum-L_2 and minimum-L_1 estimation with sample variance and interquartile range of the residuals as scale estimates under a uniform and four contaminated normal error distributions. Already under innocent looking contaminations the true mortalities may be underestimated by 50% in the long run. Moreover, the logarithmic transformation introduces an instability into the model that results in a large discrepancy between rg_Huber estimates as the tuning constant regulating the degree of robustness varies. Contrary to the logarithm the square root stabilizes variance, diminishes the influence of outliers, automatically copes with observed zeros, allows the `nonparametric' back-transformation formula E Y^2 = \mue^2 + \sigma^2, and in the homoskedastic case avoids a systematic bias of minimum-L_2 estimation with sample variance. For the company-specific table 3 of [Loeb94], in the age range of 20-65 years, we fit a parabola to root mortalities by minimum-L_2 , minimum-L_1, and robust rg_Huber regression estimates, and a cubic and exponential by least squares. The fits thus obtained in the original model are excellent and practically indistinguishable by a \chi^2 goodness-of-fit test. Finally , dispensing with the transformation of observations, we employ a Poisson generalized linear model and fit an exponential and a cubic by maximum likelihood.