Abstract
In two recent papers del Barrio et al. (1999) and del Barrio et al. (2000) consider a new class of goodness-of-fit statistics based on theL 2-Wasserstein distance. They derive the limiting distribution of these statistics and show that the normal distribution is the only location-scale family for which this limiting distribution has the “loss of degrees of freedom” property, due to the estimation of the unknown parameters. In this paper a weightedL 2-Wasserstein distance is considered and it is proven that these statistics retain the loss of degrees of freedom property for general classes of distributions if applied separately to the location family and to the scale family and if the “right” weight function is used. These weight functions are such that the corresponding minimum distance estimators for the location parameter and the scale parameter are asymptotically efficient. Examples are discussed for both location and scale families.
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de Wet, T. Goodnes-of-fit tests for location and scale families based on a weighted L2-Wasserstein distance measure. Test 11, 89–107 (2002). https://doi.org/10.1007/BF02595731
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DOI: https://doi.org/10.1007/BF02595731
Key Words
- goodness-of-fit
- Wasserstein distance
- location family
- scale family
- quantile process
- Brownian bridge
- minimum distance estimation
- limiting distributions
- asymptotic efficiency
- Karhunen-Loève expansion
- sum of weighted chisquares
- loss of degrees of freedom