The Cramér-von Mises test is one of the classical goodness-of-fit tests, for simple as well as for composite null hypotheses. In the classical theory of independent and identically distributed data the corresponding test statistic is based on the empirical distribution function, which is well known to be the nonparametric maximum likelihood estimator of the underlying distribution function if this is completely unknown.In the presence of some nonparametric auxiliary information about the underlying distribution like a known mean, however, the nonparametric maximum likelihood estimator is a modified empirical distribution function, which puts random masses on the observations in order to take the additional information into account. Substituting this modified distribution function for the classical empirical distribution function in the Cramér-von Mises statistic thus yields a modified test statistic and hence two competing Cramér-von Mises tests.In this thesis the performance of these two tests is compared by means of their limiting Pitman asymptotic relative efficiency (ARE), which describes approximately the ratio of the minimal sample sizes needed with each of the tests to obtain a given power at a small significance level and under alternatives close to the null hypothesis.For independent and identically distributed centered observations the limiting Pitman ARE of the two Cramér-von Mises tests is derived for testing a simple null hypothesis as well as for testing the composite null hypothesis that the true distribution belongs to the scale family of a generalized normal distribution.Moreover, goodness-of-fit testing for the error distribution of stable autoregressive processes with independent and identically distributed centered errors is considered. In this case the classical Cramér-von Mises statistic is based on the empirical distribution function of the residuals as an estimator for the true error distribution function. As the underlying error distribution is centered, a residual empirical distribution function that incorporates this information can be constructed analogously to the case of independent and identically distributed centered data. Hence, by replacing the classical empirical distribution function of the residuals with this modified one we get a modified version of the Cramér-von Mises statistic again. As before, the corresponding Cramér-von Mises tests are compared with the help of their limiting Pitman ARE for testing a simple null hypothesis as well as for testing the composite null hypothesis mentioned above.
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