Linear representation of M-estimates in linear models

Consider the linear regression model, y<SUB>i</SUB> = x<SUB>i</SUB>β<SUB>0</SUB> + e<SUB>i</SUB>, i = l,...,n, and an M-estimate β of β<SUB>0</SUB> obtained by minimizing ∑ρ(y<SUB>i</SUB> - x<SUB>i</SUB>β), where ρ is a convex function. Let S<SUB>n</SUB> = ∑X<SUB>i</SUB>X<SUB>i</SUB>X<SUB>i</SUB> and r<SUB>n</SUB> = S<SUB>n</SUB><SUP>1/2</SUP> (β - β<SUB>0</SUB>) - S<SUB>n</SUB><SUP>-1/2</SUP>x<SUB>i</SUB>h(e<SUB>i</SUB>), where, with a suitable choice of h(·), the expression ∑ x<SUB>i</SUB>h(e<SUB>i</SUB>) provides a linear representation of β. Bahadur (1966) obtained the order of r<SUB>n</SUB> as n → ∞ when β<SUB>0</SUB> 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