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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

##### MSC:
 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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