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Goodness of fit tests are unparametrized tests and therefore cannot directly be used to test if a test sample originates from a given distribution family. One method of solving this problem is to estimate the parameters of the distribution family before applying the test statistic on the test sample. This approach is called goodness of fit test with estimated parameters. Goodness of fit tests with estimated parameters are not a well researched field in hypothesis testing, due to their strong dependency on the explicit distribution family and selected parameter estimation. This thesis investigates problematic points of goodness of fit tests with estimated parameters, such as the incompatibility of certain parameter estimations with certain test statistics and distribution families. The thesis further provides restrictions on the parameter estimations, which guarantee the applicability of goodness of fit tests with estimated parameters on location scale distribution families. The thesis further provides an evaluation of certain test statistics for both the goodness of fit and the goodness of fit with estimated parameters test case. The investigated test statistics are the Kolmogorov-Smirnov, Cramér-von-Mises, Anderson-Darling and Zhang tests, which are compared on basis of their test powers. Three parameter estimations are investigated in combination with multiple distribution families to provide an overview of the test statistics powers. The evaluation of the test statistics includes a comparison of the ranking of test statistics in the goodness of fit test case with and without estimated parameters. This comparison indicates that the ranking of the test powers in the test case without estimated parameters cannot directly be used to derive information on the test powers in the test case with estimated parameters. |
