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Empirical likelihood for linear models. (English) Zbl 0799.62048
If one wants to test the hypothesis that independent random vectors $$Z_ 1, \dots,Z_ n$$ are identically distributed with expected value $$\mu$$ while otherwise this common distribution remains unspecified, one can use what the author calls the “empirical likelihood ratio statistic” $R(\mu) = \max \left\{ \prod^ n_{i = 1} np_ i \mid p_ i \geq 0, \quad \sum^ n_{i = 1} p_ i = 1,\;\sum^ n_{i = 1} p_ iz_ i = \mu \right\}.$ As the theorem of the paper shows, the same statistic can be used for not necessarily identically distributed $$Z$$’s with common expectation under the hypothesis, and the limit distribution of $$-2 \log R (\mu)$$ is $$\chi^ 2 (q)$$ (where $$q$$ is $$\dim \mu)$$, under some mild assumption on, vaguely speaking, nondegeneracy of the covariance matrix of the $$Z$$’s and existence of $$E \| Z_ i \|^ 4$$.
Statistics like $$R(\mu)$$ could be naturally considered under various forms of linear constraints in place of $$\sum p_ i Z_ i = \mu$$. The paper specifically considers regression models like $$Y_ i = X_ i' \beta + \varepsilon_ i$$ with mild assumptions on the distribution of the $$\varepsilon_ i$$’s. In this case the constraints take the form $\sum p_ i x_ i (y_ i - x_ i' \beta) = 0.$ The paper contains also a section on ANOVA, one called “variance modelling” and one with a numerical example.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics 62G10 Nonparametric hypothesis testing 62J05 Linear regression; mixed models 62G07 Density estimation
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