Smoothing overparameterized regression models

Abstract: Over-parameterized regression models occur throughout statistics and are often found, though not exclusively, when data are arranged in an array. Plots of fitted values for such models can suggest that smoothing is appropriate. Pe-nalized splines are a very popular method of smoothing but their use in this set-ting is not straightforward. We discuss the difficulties of using penalized splines to smooth overparameterized regression models and suggest a new smoothing paradigm which gets round these difficulties. We call our method direct smooth-ing and illustrate it by smoothing the Lee-Carter model, an over-parameterized model used in the modelling and forecasting of human mortality. We illustrate our methods with male Swedish mortality data taken from the Human Mortality Database