Optimality in weighted L₂-Wasserstein goodness-of-fit statistics

In Del Barrio, Cuesta-Albertos, Matran and Rodriguez-Rodriguez (1999) and Del Barrio, Cuesta-Albertos and Matran (2000), the authors introduced a new class of goodness-of-fit statistics based on the L₂-Wasserstein distance. It was shown that the desirable property of loss of degrees-of-freedom holds only under normality. Furthermore, these statistics have some limitations in their applicability to heavier-tailed distributions. To overcome these problems, the use of weight functions in the statistics was proposed and investigated by De Wet (2000), De Wet (2002) and Csörgő (2002). In the former the issue of loss of degrees-of-freedom was considered and in the latter the application to heavier-tailed distributions. In De Wet (2000) and De Wet (2002) it was shown how the weight functions could be chosen in order to retain the loss of degrees-of-freedom property separately for location and scale families. The weight functions that give this property, are the ones that give asymptotically optimal estimators for respectively the location and scale parameters – thus estimation optimality. In this paper we show that in the location case, this choice of “estimation optimal” weight function also gives “testing optimality”, where the latter is measured in terms of approximate Bahadur efficiencies