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

Portnoy, Stephen

Ann. Stat. 12, 1298-1309 (1984).

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

Keywords:

consistency; robustness; M-estimation; general linear model; asymptotic normality; linear combinations