International Biometric Society Asymmetric least squares smoothing and life expectancy frontiers

In the light of worldwide increasing life expectancy, demographers are interested in analyzing its limits. When studying a scatterplot of observed (time series) data such as the development of life expectancy over time and as a function of socio-economic indicators, we might not only be interested in the trend, but also in the spread around it. Quantile smoothing (QS) is an effective and popular tool for this purpose. QS is based on asymmetrically weighting the sum of absolute values of residuals. We propose asymmetrically weighting the sum of squares of residuals that leads to so called expectiles as introduced in [3]. It is extremely easy to t any asymmetric least squares (ALS) model: simply iterate between weighted regression and re-compute the weights. The goal function is convex, so a unique minimum is guaranteed. We combine ALS with P-splines as presented in [1]. As in any smoothing problem, it is important to have an automatic method to determine a good value of the smoothing parameter- in our case the weight of the penalty of P-splines. We present two methods: one is based on asymmetric cross-validation, the other adapts Schall’s EM algorithm for estimating variance components. In the asymmetric variant of the classical cross-validation score we introduce a weight vector into the calculation and perform a gri