Abstract
Consider the general linear model Y=xβ+R with Y and Rn-dimensional, βp-dimensional, and X an n×p matrix with rows x′i. Let ψ be given and let ˆβ be an M-estimator of β satisfying 0=∑xiψ(Yi−x′iˆβ). Previous authors have considered consistency and asymptotic normality of ˆβ when p is permitted to grow, but they have required at least p2/n→0. Here the following result is presented: in typical regression cases, under reasonable conditions if p(logp)/n→0 then ‖ˆβ−β‖2=Op(p/n). A subsequent paper will show that ˆβ has a normal approximation in Rp if (plogp)3/2/n→0 and that maxi|x′i(ˆβ−β)|→p0 (which would not follow from norm consistency if p2/n→∞). In ANOVA cases, ˆβ is not norm consistent, but it is shown here that max|x′i(ˆβ−β)|→p0 if plogp/n→0. A normality result for arbitrary linear combinations a′(ˆβ−β) is also presented in this case.
Citation
Stephen Portnoy. "Asymptotic Behavior of M-Estimators of p Regression Parameters when p2/n is Large. I. Consistency." Ann. Statist. 12 (4) 1298 - 1309, December, 1984. https://doi.org/10.1214/aos/1176346793
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