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On the asymptotic normality of the L 2 -Distance Class of Statistics with Estimated Parameters

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  • Kanta Naito

Abstract

In the problem of testing goodness-of-fit, widely used test statistics form the L 2 -distance. This paper studies the L 2 -distance class of statistics for testing goodness-of-fit. The phrase " L 2 -distance class" means they consistently estimate the L 2 -distance which measures the discrepancy between two probability distributions. Especially the case in which statistics include parameter estimators is investigated. It is shown that the proposed statistic has asymptotic normality under both the null and the alternative distribution. This work is essentially a generalization of the result due to Ahmad (1993) for the particular case of Cramér-von Mises statistic and is closely related to that by de Wet and Randles (1987). Several examples that illustrate the theory are also given.

Suggested Citation

  • Kanta Naito, 1997. "On the asymptotic normality of the L 2 -Distance Class of Statistics with Estimated Parameters," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 8(3), pages 199-214, September.
    • Handle: RePEc:taf:gnstxx:v:8:y:1997:i:3:p:199-214
    DOI: 10.1080/10485259708832720
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    Cited by:

    1. L. Baringhaus & B. Ebner & N. Henze, 2017. "The limit distribution of weighted $$L^2$$ L 2 -goodness-of-fit statistics under fixed alternatives, with applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 969-995, October.

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