Asymptotic comparison of Cramér-von Mises and nonparametric function estimation techniques for testing goodness-of-fit (Q1208660)

Two statistics of goodness of fit, closely related to the Cramér-von Mises (CVM) statistic \(C^ 2_ n\) and the Neyman smooth statistic \(T_{n,m}\) are derived here from the standpoint of nonparametric density estimation. It is shown that as \(n\) and \(m\) tend to infinity and \(T_{n,m}\) is recentered and rescaled appropriately, it will have a normal limiting distribution under both the null hypothesis \(H_ 0\) and Pitman type alternatives. In contrast the \(C^ 2_ n\) statistic has nontrivial power against alternatives approaching the null hypothesis as fast as \(1/\sqrt n\). Some small scale simulation results are also reported which demonstrate that the asymptotics extend in principle to the case of finite samples.