Linear representation of M-estimates in linear models

Abstract

Consider the linear regression model, y_i = x_iβ₀ + e_i, i = l,...,n, and an M-estimate β of β₀ obtained by minimizing ∑ρ(y_i - x_iβ), where ρ is a convex function. Let S_n = ∑X_iX_iX_i and r_n = S_n^1/2 (β - β₀) - S_n^-1/2x_ih(e_i), where, with a suitable choice of h(·), the expression ∑ x_ih(e_i) provides a linear representation of β. Bahadur (1966) obtained the order of r_n as n → ∞ when β₀ is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of rn as n → ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.