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Asymptotic behavior of M estimators of p regression parameters when $$p^ 2/n$$ is large. II: Normal approximation. (English) Zbl 0601.62026
[For part I see ibid. 12, 1298-1309 (1984; Zbl 0584.62050).]
The M-estimator $${\hat \beta}$$ of the parameters in the general linear model $$Y_ i=\sum^{p}_{j=1}\beta_ jx_{ij}+R_ i$$ is defined by the system of equations $\sum^{n}_{i=1}x_{ij}\Psi (Y_ i- \sum^{p}_{j=1}\beta_ jx_{ij})=0,\quad j=1,...,p,$ where $$\Psi$$ : $$R\to R$$. In a companion paper and in this paper the author considers asymptotic properties of $${\hat \beta}$$. In the present part of the investigation it is shown that (i) if $$p^{3/2}(\log n)/n\to 0$$ then $$\max | x_ i'({\hat \beta}-\beta)|^ p\to 0$$, and (ii) if (p log n)$${}^{3/2}\to 0$$ then for any sequence $$\{a_ n\}$$ with $$a_ n\in R^ p$$, $$a_ n'({\hat \beta}-\beta)$$ converges in distribution to a normal distribution and (under stronger conditions) a uniform normal approximation for the distribution of $${\hat \beta}$$ will hold which yields a $$\chi^ 2_ p$$ approximation for ($${\hat \beta}$$- $$\beta)$$’(X’X)($${\hat \beta}$$-$$\beta)$$, X being the $$n\times p$$ matrix with elements $$x_{ij}$$ and rows $$x_ i$$.
Reviewer: H.Nyquist

##### MSC:
 62E20 Asymptotic distribution theory in statistics 62F35 Robustness and adaptive procedures (parametric inference) 62J05 Linear regression; mixed models
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