Estimation of Mortalities

If a linear regression is t to log transformed mortalities and the estimate is back transformed according to the formula E e e 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 and minimum L esti mation 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 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 in uence of outliers automatically copes with observed zeros allows the nonparametric back transformation formula E Y and in the homoskedastic case avoids a systematic bias of minimum L estimation with sample variance For the company speci c table of Loeb in the age range of years we t a parabola to root mortalities by minimum L minimum L and robust rg Huber regression estimates and a cubic and exponential by least squares The ts thus obtained in the original model are excellent and practically indistinguishable by a goodness of t test Finally dispensing with the transformation of observations we employ a Poisson generalized linear model and t an exponential and a cubic by maximum likelihood Part of the work was done while the author was visiting the Statistics Department University of Munich