Scale measures for bandwidth selection

Both high tech and also quick and dirty methods for selecting the bandwidth of a kernel density estimator are usually based on measures of scale. An example is given to illustrate how the usual scale measures can be arbitrarily bad. This motivates development of several improved scale measures. These are assessed through simulation, and experience with a real data set. Our best scale measure works at least as well as all other proposals in nearly all settings, and is much better quite often.