## 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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### Cite this article

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