Data driven smooth tests for composite hypotheses comparison of powers

Rao's score statistic is a standard tool for constructing statistical tests.If departures from the null model are described by some k-dimensional exponential family the resulting score test is called also smooth test or Neyman's smooth test with k components. An important practical question in applying a smooth test in the goodness-of-fit problem is how large k should be taken. Since a wrong choice may give a considerable loss of power,it is important to make a careful selection.Renewed research in this area shows that the simple question has no simple deterministic answer. Therefore,edwina introduced,for testing a simple goodness-of-fit hypothesis,a data driven version of Neyman's smooth test. First,Schwarz's rule is applied to find a suitable dimension and then the smooth test statistic in the “right" dimension finishes the job. Simulation results and some theoretical considerations show that this data driven version of smooth tests performs well for a wide range of alternatives,and is competitive with other recently introduced (data driven) procedures.This data-dependent choice of the number of components is extended in this paper to testing the goodness-of-fit problem with composite null hypothesis,being of more practical interest.A k-dimensional exponential family for modelling departures from the null hypothesis is given and the related Rao's score test is described. A suitable version of Schwarz's rule is proposed and some simplifications of it are discussed.To check validity of the proposed construction,the method is applied to testing exponentiality and normality.In the extensive simulation study,presented in this paper,it turns out that the data driven version of smooth tests compares well for a wide range of alternatives with other,more specialized,commonly used tests