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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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