Bandwidth selection for the smoothing of distribution functions

Several approaches can be made to the choice of bandwidth in the kernel smo othing of distribution functions. Recent proposals by Sarda (1993) and by A ltman & Leger (1995) are analogues of the 'leave-one-out' and 'plug-in' met hods which have been widely used in density estimation. In contrast, a meth od of crossvalidation appropriate to the smoothing of distribution function s is proposed. Selection of the bandwidth parameter is based on unbiased es timation of a mean integrated squared error curve whose minimising value de fines an optimal smoothing parameter. This procedure is shown to lead to as ymptotically optimal bandwidth choice, not just in the usual first-order se nse but also in the second-order sense in which kernel methods improve on t he standard empirical distribution function. Some general theory on the per formance of optimal, data-based methods of bandwidth choice is also provide d, leading to results which do not have analogues in the context of density estimation. The numerical performances of all the methods discussed in the paper are compared. A bandwidth based on a simple reference distribution i s also included. Simulations suggest that the crossvalidatory proposal work s well, although the simple reference bandwidth is also quite effective.