Expectile smoothing: new perspectives on asymmetric least squares. An application to life expectancy

While initially motivated from a demographic application, this thesis develops methodology for expectile estimation. To this end first the basic model for expectile curves using least asymmetrically weighted squares (LAWS) was introduced as well as methods for smoothing in this context. The simple LAWS model was successfully applied to the original demographic data problem. Expectile curves from the simple LAWS model can cross. This is an undesirable, theoretically impossible, yet practically encountered phenomenon when estimating expectiles separately. In addition to the considerable (quantile) literature about this problem, the expectile bundle model is proposed to avoid crossings. Smooth non-crossing expectile curves are the result of this location-scale model. All curves are estimated simultaneously. Another side product is the possibility to estimate a single conditional density of the data. However, it is also possible to estimate a conditional (local) density with any set of estimated expectiles -- irrespective of the estimation method. A location-scale model is an attractive approach when supported by the data. Empirical data cannot always be modeled with this set-up. Therefore another application of asymmetric least squares, an expectile sheet, is introduced here. It also estimates a set of expectile curves simultaneously without the use of a restrictive assumption such as the expectile bundle model. A set of expectile curves can always be interpreted as a surface over the independent variable x and asymmetry p. By proceeding this way the surface is following from separate expectile curves. An expectile sheet is the direct estimation of this surface over the domain (x,p). It is based on a tensor product of B-splines covering both dimensions. Smoothing is applied in both directions. p-expectile curves are the contours of the expectile sheet for fixed p and along x. LAWS is based on iteratively reweighted least squares with asymmetric weights that are recalculated in each step. The expectile algorithm can be easily modified to estimate smooth quantile curves by dividing the original weights by the absolute value of the residuals. The new weights are updated in each iteration step. Each of the described expectile methods can be adapted to estimate quantiles by using the new weights. As an example for this adaptation of the weights quantile sheets are finally proposed in this thesis. They are based on expectile sheets with this modified weight vector