A weighted Cramer-Von Mises statistic, with some applications to clinical trials

A new weighted Cramer-von Mises statistic is proposed, in which the greatest weight is attached to the "early" observations. The proposed statistic is broadly similar to the Anderson-Darling and Rodriguez-Viollaz statistics, and the asymptotic properties are proved using an eigenfunction expansion involving Bessel functions. Practical aspects are discussed, and there is an example involving clinical trials. An advantage of our proposed test is that trials may often be stopped early because rejection of the null hypothesis is certain.