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Bootstrapping \(M\)-estimators of a multiple linear regression parameter. (English) Zbl 0792.62058

This paper considers a multiple linear regression model \(Y_ i= X_ i' \beta+\varepsilon_ i\), where the errors \(\varepsilon_ i\) are independent random variables with common distribution \(F\) and the \(X_ i\) are known design vectors. Let \(\overline{\beta}_ n\) be the \(M\)- estimator of \(\beta\) corresponding to a score function \(\psi\). Two-term Edgeworth expansions for the distributions of standardized and Studentized \(\overline{\beta}_ n\) are obtained under some conditions on \(F\), \(\psi\) and the design. Also, it is shown that the bootstrap method is second order correct.

MSC:

62J05 Linear regression; mixed models
62G09 Nonparametric statistical resampling methods
62F10 Point estimation
62E20 Asymptotic distribution theory in statistics
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