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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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