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Cramér–von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation

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  • L. Baringhaus
  • N. Henze

Abstract

We give a probabilistic interpretation of the Cramér–von Mises distance $ \Delta (F,F_0) = \int (F-F_0)^2\,{\rm d}F_0 $ Δ(F,F0)=∫(F−F0)2dF0 between continuous distribution functions F and $ F_0 $ F0. If F is unknown, we construct an asymptotic confidence interval for $ \Delta (F,F_0) $ Δ(F,F0) based on a random sample from F. Moreover, for given $ F_0 $ F0 and some value $ \Delta _0>0 $ Δ0>0, we propose an asymptotic equivalence test of the hypothesis that $ \Delta (F,F_0) \ge \Delta _0 $ Δ(F,F0)≥Δ0 against the alternative $ \Delta (F,F_0) < \Delta _0 $ Δ(F,F0)<Δ0. If such a ‘neighbourhood-of- $ F_0 $ F0 validation test’, carried out at a small asymptotic level, rejects the hypothesis, there is evidence that F is within a distance $ \Delta _0 $ Δ0 of $ F_0 $ F0. As a neighbourhood-of-exponentiality test shows, the method may be extended to the case that $ H_0 $ H0 is composite.

Suggested Citation

  • L. Baringhaus & N. Henze, 2017. "Cramér–von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 167-188, April.
    • Handle: RePEc:taf:gnstxx:v:29:y:2017:i:2:p:167-188
    DOI: 10.1080/10485252.2017.1285029
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    1. Ludwig Baringhaus & Norbert Henze, 1991. "A class of consistent tests for exponentiality based on the empirical Laplace transform," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(3), pages 551-564, September.
    2. Freitag, Gudrun & Munk, Axel, 2005. "On Hadamard differentiability in k-sample semiparametric models--with applications to the assessment of structural relationships," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 123-158, May.
    3. Axel Munk & Claudia Czado, 1998. "Nonparametric validation of similar distributions and assessment of goodness of fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 223-241.
    4. Norbert Henze & Simos G. Meintanis, 2005. "Recent and classical tests for exponentiality: a partial review with comparisons," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 61(1), pages 29-45, February.
    5. Holger Dette & Axel Munk, 2003. "Some Methodological Aspects of Validation of Models in Nonparametric Regression," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(2), pages 207-244, May.
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    Cited by:

    1. Steffen Betsch & Bruno Ebner, 2020. "Testing normality via a distributional fixed point property in the Stein characterization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 105-138, March.
    2. L. Baringhaus & D. Gaigall & J. P. Thiele, 2018. "Statistical inference for $$L^2$$ L 2 -distances to uniformity," Computational Statistics, Springer, vol. 33(4), pages 1863-1896, December.
    3. Steffen Betsch & Bruno Ebner, 2019. "A new characterization of the Gamma distribution and associated goodness-of-fit tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(7), pages 779-806, October.
    4. Alessandro Barbiero & Asmerilda Hitaj, 2023. "Discrete approximations of continuous probability distributions obtained by minimizing Cramér-von Mises-type distances," Statistical Papers, Springer, vol. 64(5), pages 1669-1697, October.
    5. Neil Shephard, 2020. "An estimator for predictive regression: reliable inference for financial economics," Papers 2008.06130, arXiv.org.

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