On the automatic selection of the tuning parameter appearing in certain families of goodness-of-fit tests

The situation, common in the current literature, is that of a whole family of location-scale/scale invariant test statistics, indexed by a parameter , is available to test the goodness of fit of F, the underlying distribution function of a set of independent real-valued random variables, to a location-scale/scale family of distribution functions. The power properties of the tests associated with the different statistics usually depend on the parameter , called the tuning parameter, which is the reason that its choice is crucial to obtain a performing test procedure. In this paper, we address the automatic selection of the tuning parameter when is finite, as well as the calibration of the associated goodness-of-fit test procedure. Examples of existing and new tuning parameter selectors are discussed, and the methodology presented of combining different test statistics into a single test procedure is applied to well known families of test statistics for normality and exponentiality. A simulation study is carried out to access the power of the different tests under consideration, and to compare them with the fixed tuning parameter procedure, usually recommended in the literature.