Asymptotic Behavior of M-Estimators of p Regression Parameters when p²/n is Large. I. Consistency

Consider the general linear model Y = x + R with Y and R n-dimensional, p-dimensional, and X an n p matrix with rows x'ᵢ. Let be given and let be an M-estimator of satisfying 0 = xᵢ (Yᵢ - x'ᵢ). Previous authors have considered consistency and asymptotic normality of 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 (p) /n 0 then \| - \|² = Oₚ (p/n). A subsequent paper will show that has a normal approximation in Rᵖ if (p p) ^3/2/n 0 and that ᵢ|x'ᵢ (-) | ₚ 0 (which would not follow from norm consistency if p²/n). In ANOVA cases, is not norm consistent, but it is shown here that |x'ᵢ (-) | ₚ 0 if p p/n 0. A normality result for arbitrary linear combinations a' (-) is also presented in this case.