An approximation to the null distribution of a class of Cramér–von Mises statistics

Abstract

A class of goodness-of-fit tests of the Cramér–von Mises type is considered. More specifically, the test statistic of each test is an L2-norm of the difference between the empirical characteristic function associated with a random sample and a parametric estimator of the characteristic function of the population in the null hypothesis. The null distribution of these statistics is unknown and we study a way of estimating it, which is based on approximating the asymptotic null distribution. The asymptotic null distribution is a linear combination of independent chi-squared variates, where the weights are the eigenvalues of certain operator. The calculation of these eigenvalues is, in most cases, a very difficult task. In order to bypass this computation we approximate the test statistic. The asymptotic null distribution of the approximation is again a linear combination of chi-squared variates, but now the weights can be easily approximated. A simulation study is carried out to examine the accuracy of the proposed approximation for finite sample sizes. Although we center our attention on the aforementioned class, the methodology exposed can be applied for approximating the null distribution of other Cramér–von Mises type test statistics.