The Maximal Smoothing Principle in Density Estimation

We propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators. It has come to be realized in recent years (e.g., see Hall and Marron 1987; Scott and Terrell 1987) that cross-validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value. Their sampling variability is simply too large. The alternative discussed here, the maximal smoothing principle, suggests that we consider using the most smoothing that is consistent with the estimated scale of our data. This greatly generalizes and exploits a phenomenon noted in Terrell and Scott (1985), that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons. The method avoids the extreme sampling variability of cross-validation by using ordinary scale estimators such as the standard deviation and interquartile range, which have order n −1 variability; cross-validated parameters have orders of variability such as n −1/5. The disadvantage is that maximal smoothing parameters are conservative, rather than asymptotically optimal. Because they tend to lose information, they should be used in conjunction with other data displays that retain more of the features of the original sample. On the other hand, such conservative methods are widely valued by statisticians because they discourage naive overinterpretation of one's data. Maximal smoothing parameters are here derived for histograms and kernel methods, using not only the standard deviation but several more resistant methods of scale estimation. The method is then applied to density estimation on the half-line, on finite intervals, and in several variables.