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The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications

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Abstract

We present a general result on the limit distribution of weighted one- and two-sample \(L^2\)-goodness-of-fit test statistics of some hypothesis \(H_0\) under fixed alternatives. Applications include an approximation of the power function of such tests, asymptotic confidence intervals of the distance of an underlying distribution with respect to the distributions under \(H_0\), and an asymptotic equivalence test that is able to validate certain neighborhoods of \(H_0\).

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Acknowledgments

The authors would like to thank Ya. Yu. Nikitin for drawing our attention to the paper of Chapman (1958) and B. Klar for pointing out the reference Naito (1997). Thanks go to two anonymous referees for their careful reading of the manuscript and for helpful suggestions.

Author information

Correspondence to B. Ebner.

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Baringhaus, L., Ebner, B. & Henze, N. The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications. Ann Inst Stat Math 69, 969–995 (2017). https://doi.org/10.1007/s10463-016-0567-8

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Keywords

  • Goodness-of-fit test
  • Weighted \(L^2\)-statistic
  • Fixed alternative
  • Empirical transform
  • Asymptotic equivalence test