Analysis of Theoretical and Empirical Measures of Fitting and Smoothing for Restricted Function Estimators

Following Henderson (1916) who developed a smoothing measure as a function of the weight system of a linear filter, Dagum and Luati (2002) proposed a set of local statistical measures of bias, variance and mean square error which are intrinsic to the smoother and, thus, independent of the data to which they will be applied on. Theoretical measures of local fitting and smoothing are calculated on the basis of the weight systems of the following nonparametric function estimators, Loess of degree 1 and 2, the cubic smoothing spline, the Gaussian kernel, and the 13-term Henderson filter. Our aim is to evaluate the extent to which these local or weight-based measures of fitting and smoothing can be used to obtain a priori general information on the global (data-based) goodness of fit and smoothness when such filters are applied to real time series. For each function estimator, we calculate global measures of fitting and smoothing using two large samples of real and simulated series characterized by different degrees of variability. The results show that the theoretical (weight-based) local smoothing measures are always in agreement with the global empirical ones. Similarly, the local (weight-based) mean square error, analyzed in terms of bias and variance composition, provides sound a priori information on the global goodness of fit given by the symmetric filters of each nonparametric estimator. For the asymmetric filters, the above analysis must be done taking into consideration also the impact of phase shifts which can be inferred from the smoothing measures.