INTERMEDIATE APPROACH TO COMPARISON OF SOME GOODNESS-OF-FIT TESTS

TADEUSZ INGLOT^{1} AND TERESA LEDWINA^{2}

1Institute of Mathematics, Polish Academy of Sciences and Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

2Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617 Wrocław, Poland

(Received October 5, 1999; revised March 21, 2000)

Abstract. In this paper we present the intermediate approach to investigating asymptotic power and measuring the efficiency of nonparametric goodness-of-fit tests for testing uniformity. Contrary to the classical Pitman approach, the intermediate approach allows the explicit quantitative comparison of powers and calculation of efficiencies. For standard tests, like the Cramér-von Mises test, an intermediate approach gives conclusions consistent with qualitative results obtained using the Pitman approach. For other more complicated cases the Pitman approach does not give the right picture of power behaviour. An example is the data driven Neyman test we present in this paper. In this case the intermediate approach gives results consistent with finite sample results. Moreover, using this setting, we prove that the data driven Neyman test is asymptotically the most powerful and efficient under any smooth departures from uniformity. This result shows that, contrary to classical tests being efficient and the most powerful under one particular type of departure from uniformity, the new test is an adaptive one.

Key words and phrases: Intermediate efficiency, vanishing shortcoming, goodnessof-fit test, data driven test, smooth test, Schwarz rule.