Asymptotic behavior of M-estimators of p regression parameters when \(p^ 2/n\) is large. I. Consistency. (English) Zbl 0584.62050

M-estimation of the regression parameters in the general linear model \(Y_ i=\sum^{p}_{j=1}\beta_ jx_{ji}+R_ i\) is defined as the solution to the system of equations \[ \sum^{n}_{i=1}x_{ji}\psi (Y_ i-\sum^{p}_{j=1}\beta_ jx_{ji}),\quad j=1,...,p. \] This paper considers asymptotic properties of M-estimators, \({\hat \beta}\).
In the case of linear regression it is shown that if \(\psi\) is increasing, p(log p)/n\(\to 0\), and some other relatively mild conditions hold, then \(\| {\hat \beta}\|^ 2=O_ p(p/n)\). In the analysis of variance case of the general linear model it is shown that if p(log p)/n\(\to 0\) then at least \(\max_ j| {\hat \beta}_ j| =O_ p((p(\log p)/n)^{1/2})\). Also a result giving asymptotic normality for arbitrary linear combinations a’\({\hat \beta}\) is presented.
Reviewer: H.Nyquist


62F35 Robustness and adaptive procedures (parametric inference)
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
62E20 Asymptotic distribution theory in statistics
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