Abstract
This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramér-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.
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Dedicated to Miguel Martín Díaz whose scientific criticism has greatly inspirated our research by years.
Research partially supported by DGICYT, grants PB98-0369-C02-01 and 02. E. del barrio and C. Matrán have also been supproted by PAPIJCL grant VA08/97.
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del Barrio, E., Cuesta-Albertos, J.A., Matrán, C. et al. Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests. Test 9, 1–96 (2000). https://doi.org/10.1007/BF02595852
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DOI: https://doi.org/10.1007/BF02595852
Key words
- Goodness-of-fit
- correlation tests
- Cramér-von Mises
- empirical and quantile/processes
- Kolmogorov-Smirnov
- Shapiro-Wilk
- strong approximations
- Wasserstein distance