Asymptotic optimality of data driven smooth tests for location-scale family

SUMMARY. Data driven smooth tests have been introduced to enlarge range of sensitivity of classical goodness of fit tests. In case of simple goodness of fit hypothesis there is an evidence that this goal has been in some sense achieved. This paper aims to prove that also in case of presence of nuisance parameteres the construction meets some optimal properties. The tool we use to show it is immediate local comparison of powers of data driven smooth tests and the best possible test under given alternative. We prove that the difference of powers vanishes as sample size increases. This shows that data driven tests are optimal ones in a very natural sense. 1